File "complex-numbers-roots-of-polynomials.html"
Path: /StudyIB/mathsanalysis/page/675/complex-numbers-roots-of-polynomialshtml
File size: 60.48 KB
MIME-type: text/html
Charset: utf-8
<!DOCTYPE html><html lang="EN"><head> <script>(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer','GTM-PZBWZ8J');</script> <title>Complex Numbers - Roots of Polynomials</title><!-- Removed by WebCopy --><!--<base href="/">--><!-- Removed by WebCopy --><meta http-equiv="x-ua-compatible" content="IE=Edge"><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1"><link href="../../../css/bootstrap.flatly.min.css" rel="stylesheet" media="screen"><link rel="stylesheet" href="../../../fonts/awesome/css/font-awesome.min.css"><link href="../../../js/jquery-fancybox/jquery.fancybox.min.css" type="text/css" rel="stylesheet"><link rel="stylesheet" href="../../../css/style.min.css?v=202301301645"><meta name="robots" content="index, follow"><meta name="googlebot" content="noarchive"><meta prefix="og: http://ogp.me/ns#" property="og:title" name="og:title" content="StudyIB Maths: Analysis & Approaches: Complex Numbers - Roots of Polynomials"> <meta prefix="og: http://ogp.me/ns#" property="og:image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta prefix="og: http://ogp.me/ns#" property="og:description" name="og:description" content="The Conjugate Root Theorem states that if the complex number a + ib is a root of a polynomial in one variable with real coefficients, then the complex conjugate a - bi also a root of that polynomial. This is a useful theorem for solving polynomials with re"> <meta prefix="og: http://ogp.me/ns#" property="og:url" name="og:url" content="https://studyib.net/mathsanalysis/page/675/complex-numbers-roots-of-polynomials"> <meta prefix="og: http://ogp.me/ns#" property="og:site_name" name="og:site_name" content="StudyIB - Your IB learning companion"> <meta prefix="og: http://ogp.me/ns#" property="og:locale" name="og:locale" content="en"> <meta prefix="og: http://ogp.me/ns#" property="og:type" name="og:type" content="website"> <meta name="description" content="The Conjugate Root Theorem states that if the complex number a + ib is a root of a polynomial in one variable with real coefficients, then the complex conjugate a - bi also a root of that polynomial. This is a useful theorem for solving polynomials with re"> <meta name="image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="keywords" content="Conjugate root, Complex roots, Cubic, IB, IBDP, InThinking, International Baccalaureate, Revision, Revision websites, Student Sites, Students, Learning, Guidance, Exam, Questions, Practice problems, Diagnose, Help, Notes"> <meta itemprop="name" content="StudyIB Maths: Analysis & Approaches: Complex Numbers - Roots of Polynomials"> <meta itemprop="description" content="The Conjugate Root Theorem states that if the complex number a + ib is a root of a polynomial in one variable with real coefficients, then the complex conjugate a - bi also a root of that polynomial. This is a useful theorem for solving polynomials with re"> <meta itemprop="image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:title" content="StudyIB Maths: Analysis & Approaches: Complex Numbers - Roots of Polynomials"> <meta name="twitter:description" content="The Conjugate Root Theorem states that if the complex number a + ib is a root of a polynomial in one variable with real coefficients, then the complex conjugate a - bi also a root of that polynomial. This is a useful theorem for solving polynomials with re"> <meta name="twitter:image" content="https://studyib.net/img/studyib-card-default.jpg"> <meta name="twitter:creator" content="@inthinker"><link rel="stylesheet" href="../../../css/snippets.min.css?v=202209111300"><link rel="stylesheet" href="../../../css/article.min.css?v=202211221000"><link rel="stylesheet" type="text/css" href="../../../js/slick-carousel/slick.min.css"><link rel="stylesheet" type="text/css" href="../../../js/slick-carousel/slick-theme.min.css"><style type="text/css">.filter-sl-only { display: none; }</style><script src="../../../ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script><script src="../../../js/ifvisible.min.js"></script><script>ifvisible.setIdleDuration(300);</script><link rel="apple-touch-icon-precomposed" sizes="57x57" href="../../../img/favicon/apple-touch-icon-57x57.png"> <link rel="apple-touch-icon-precomposed" sizes="114x114" href="../../../img/favicon/apple-touch-icon-114x114.png"> <link rel="apple-touch-icon-precomposed" sizes="72x72" href="../../../img/favicon/apple-touch-icon-72x72.png"> <link rel="apple-touch-icon-precomposed" sizes="144x144" href="../../../img/favicon/apple-touch-icon-144x144.png"> <link rel="apple-touch-icon-precomposed" sizes="60x60" href="../../../img/favicon/apple-touch-icon-60x60.png"> <link rel="apple-touch-icon-precomposed" sizes="120x120" href="../../../img/favicon/apple-touch-icon-120x120.png"> <link rel="apple-touch-icon-precomposed" sizes="76x76" href="../../../img/favicon/apple-touch-icon-76x76.png"> <link rel="apple-touch-icon-precomposed" sizes="152x152" href="../../../img/favicon/apple-touch-icon-152x152.png"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-196x196.png" sizes="196x196"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-96x96.png" sizes="96x96"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-32x32.png" sizes="32x32"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-16x16.png" sizes="16x16"> <link rel="icon" type="image/png" href="../../../img/favicon/favicon-128.png" sizes="128x128"> <meta name="application-name" content="StudyIB: Your IB Learning Companion"> <meta name="msapplication-TileColor" content="#A5BED5"> <meta name="msapplication-TileImage" content="/img/favicon/mstile-144x144.png"> <meta name="msapplication-square70x70logo" content="/img/favicon/mstile-70x70.png"> <meta name="msapplication-square150x150logo" content="/img/favicon/mstile-150x150.png"> <meta name="msapplication-wide310x150logo" content="/img/favicon/mstile-310x150.png"> <meta name="msapplication-square310x310logo" content="/img/favicon/mstile-310x310.png"><script>var stdHash = "6da675cbcb3d25f050b05557349abc79", stdTicket = "082b9c9c4ae3624d";</script><script src="../../../js/user/local-stats.min.js?v=202205311700"></script><link href="../../../css/subjects-frontpage.min.css?v=202302031000" rel="stylesheet"></head><body class="public mathsanalysis"> <noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-PZBWZ8J" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript> <div id="top-header"> <div class="wmap general hidden-sm hidden-xs subjects"> <div class="layout-wrapper"> <div class="container-fluid"> <a href=""> <img class="header-thinker" src="../../../img/header-thinker.svg"> </a> <h1> <a href="../../../mathsanalysis.html"> <span style="font-family: 'Helvetica Narrow','Arial Narrow',Tahoma,Arial,Helvetica,sans-serif;font-stretch: condensed;letter-spacing: -1px">IBDP Maths: Analysis & Approaches</span> </a> <a href="https://inthinking.net" title="inthinking.net" class="inthinking-logo pull-right"> <img src="../../../img/header-logo.svg" style="height: 80px; width: auto;"> </a> </h1> <p class="slogan">InThinking Revision Sites for students</p> <p class="author">Website by <strong>Richard Wade</strong></p> <p class="updated">Updated 3 February 2023</p> </div> </div> </div> <nav id="public-topnav" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <a class="navbar-brand hidden-md hidden-lg" href="../../../index.htm"> <img class="header-xs-thinker" src="../../../img/header-thinker.svg"> </a> <button type="button" class="collapsed navbar-toggle" data-toggle="collapse" data-target="#subject-navbar-collapse" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <i class="fa fa-fw fa-2x fa-user"></i> </button> <div class="brand hidden-lg hidden-md hidden-sm"> <a class="brand-xs" href="../../../mathsanalysis.html"> <strong class="title" style="white-space: nowrap;font-size: 18px;">Maths: A&A</strong> </a> </div> </div> <div class="collapse navbar-collapse" id="subject-navbar-collapse"> <ul class="nav navbar-bar navbar-userbox hidden-md hidden-lg"><li class="dropdown"><a href="#" class="dropdown-toggle" data-toggle="dropdown"><i class="fa fa-fw fa-lg fa-user-circle" style="margin-right: 5px;"></i><span style="font-size: 16px;"></span></a><ul class="dropdown-menu"><li class="dropdown-submenu"><a class="dropdown-toggle" href="#" data-toggle="dropdown"><i class="fa fa-fw fa-globe"></i> Subjects</a><ul class="dropdown-menu"><li></li><li class=""><a href="../../../biology.html">DP Biology</a></li><li class=""><a href="../../../chemistry.html">DP Chemistry</a></li><li class=""><a href="../../../englishalanglit.html">DP English A: Language & Literature</a></li><li class="active"><a href="../../../mathsanalysis.html"><i class="fa fa-caret-right"></i> DP Maths: Analysis & Approaches</a></li><li class=""><a href="../../../mathsapplications.html">DP Maths: Applications & Interpretations SL</a></li><li class=""><a href="../../../physics.html">DP Physics</a></li><li class=""><a href="../../../spanishb.html">DP Spanish B</a></li></ul></li><li class="menu-item"><a href="../../../user.html"><i class="fa fa-fw fa-dashboard"></i> Dashboard</a></li><li class="menu-item"><a href="../../../user/profile.html"><i class="fa fa-fw fa-cog"></i> My profile</a></li><li class="menu-item"><a href="../../../user/messages.html"><i class="fa fa-fw fa-envelope"></i> Messages</a></li><li class="menu-item hidden-md hidden-lg mt-xs-3"><a href="../../../index.htm?logout=1" class="btn btn-primary btn-xs-block"><i class="fa fa-fw fa-lg fa-power-off text-danger"></i>Log out</a></li></ul></li></ul> <ul class="nav navbar-bar"><li class=""><a class="home" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-home"></i> Home</a></li><li class=""><a class="topics" href="#"><i class="fa fa-fw fa-th-large"></i> Topics</a></li><li class=""><a class="favorites" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-star"></i> My favorites</a></li><li class=""><a class="qbank" href="../../test-your-knowledge.html"><i class="fa fa-fw fa-question"></i> Question bank</a></li><li class=""><a class="sitemap" href="../../sitemap.html"><i class="fa fa-fw fa-sitemap"></i> Sitemap</a></li><li class=""><a class="activity" href="../../../mathsanalysis.html"><i class="fa fa-fw fa-calendar-check-o"></i> My activity</a></li></ul> <ul class="nav navbar-bar navbar-right"> <li> <form class="navbar-form hidden-md hidden-lg" role="search" method="get" action="mathsanalysis/search"> <div class="input-group"> <input class="form-control nav-search" name="glob" type="search" placeholder="Search..."> <span class="input-group-btn"> <button type="submit" class="btn bg-blue"> <i class="fa fa-search"> </i> </button> </span> </div> </form> <a href="#" class="toggle-menu-search hidden-sm hidden-xs" data-toggle="dropdown"> <i class="fa fa-lg fa-search"></i> </a> </li></ul> </div> </div></nav><nav id="nav-menu-search" class="shadow-bottom hidden-sm hidden-xs" style="display: none;"> <div class="container-fluid"> <form class="" role="search" method="get" action="mathsanalysis/search"> <input class="form-control nav-search" name="glob" type="search" placeholder="Search Maths: A&A..."> <button class="btn btn-sm btn-primary" type="submit"> Search </button> </form> </div></nav><div class="modal fade" tabindex="-1" role="dialog" id="modal-mini-login"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-body"> <button aria-hidden="true" data-dismiss="modal" class="close hidden-xs login-button-close" type="button">×</button> <form method="post"> <div class="form-group"> <input name="user-email" class="form-control" type="text" placeholder="Email"> </div> <div class="form-group"> <input name="user-password" class="form-control" type="password" placeholder="Password"> <input name="user-fp" class="fp" value="d8c893da5db7501669ffeeac46273ba6" type="hidden"> </div> <div class="form-group"> <a href="../../../reset-password.html" class="reset-password" style="font-weight: normal;"> Forgot your password? </a> </div> <div class="form-group"> <button type="submit" name="submit-login" value="1" class="btn btn-primary btn-block text-center"> <i class="fa fa-user fa-fw"></i> Log in </button> </div> </form> </div> </div> </div></div></div><div class="modal fade" id="show-topics" tabindex="-1" role="dialog"><div id="dialog-topics" class="modal-dialog modal-dialog-topics"><div class="modal-content"><div class="modal-body modal-body-topics"><div id="new-frontpage-toplevels-topics" class="frontpage-box"><div class="row"><div class="col-md-4 col-sm-4"><div class="item"><a href="../3016/free-access-weekend.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Free Access Weekend</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/free.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2696/start-here.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Start Here</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/start-here.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2902/examination-questions.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Examination Questions</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/exam.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../2563/question-bank.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Question Bank</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/questionbank.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../537/algebra.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Algebra</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/algebra.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../539/functions.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Functions</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/function_s.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../540/geometry-trigonometry.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Geometry & Trigonometry</h3></div><div class="body"><div class="cropper" style="background-image: url('../../images/photo-by-a-href=httpsunsplash.com@pawel_czerwinskiutm_source=unsplash-utm_medium=referral-utm_content=creditcopytextpawel-czerwinskia-on-a-href=httpsunsplash.comutm_source=unsplash-utm_medium=referral-utm_co.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../549/stats-probability.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Stats & Probability</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/probability.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../550/calculus.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Calculus</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/images/calculus1.jpg'); height: 10vw;"></div></div></a></div></div><div class="col-md-4 col-sm-4"><div class="item"><a href="../857/exam-tips.html"><div class="header" style="height: 35px;"><h3 class="title special" style="margin-top: 0;">Exam Tips</h3></div><div class="body"><div class="cropper" style="background-image: url('/media/mathsanalysis/files/exam-tips/exam-tips.jpg'); height: 10vw;"></div></div></a></div></div></div></div></div></div></div></div><div class="container-fluid shadow-ith"><div class="row frontpage"> <div class="col-md-3" id="left-column"> <p class="visible-xs-block"></p><div class="dropdown user-menu hidden-xs" style="margin-bottom: 20px;"> <button type="button" class="btn btn-default dropdown-toggle" data-toggle="dropdown" style="border-radius: 0; padding: 6px 18px;"> <i class="fa fa-fw fa-user-circle"></i> <small><i class="fa fa-caret-down"></i></small> </button> <ul class="dropdown-menu"> <li class="dropdown-submenu"><a href="#"><i class="fa fa-fw fa-globe"></i> Subjects</a><ul id="user-subjects-menu" class="dropdown-menu"><li></li><li class=""><a href="../../../biology.html">DP Biology</a></li><li class=""><a href="../../../chemistry.html">DP Chemistry</a></li><li class=""><a href="../../../englishalanglit.html">DP English A: Language & Literature</a></li><li class="active"><a href="../../../mathsanalysis.html"><i class="fa fa-caret-right"></i> DP Maths: Analysis & Approaches</a></li><li class=""><a href="../../../mathsapplications.html">DP Maths: Applications & Interpretations SL</a></li><li class=""><a href="../../../physics.html">DP Physics</a></li><li class=""><a href="../../../spanishb.html">DP Spanish B</a></li></ul></li><li class="divider"></li><li><a href="../../../user.html"><i class="fa fa-fw fa-dashboard"></i> Dashboard</a></li><li><a href="../../../user/profile.html"><i class="fa fa-fw fa-cog"></i> My profile</a></li><li><a href="../../../user/messages.html"><i class="fa fa-fw fa-envelope"></i> Messages</a></li> <li class="divider"></li> <li> <a href="../../../index.htm?logout=1"> <i class="fa fa-fw fa-power-off"></i> Log out </a> </li> </ul></div> <div id="side-nav"><h4 class="side-nav-title dropdown"><a href="../537/algebra.html" class="dropdown-toggle toplevel-dropdown" data-toggle="dropdown"><i class="fa fa-ellipsis-v"></i></a> <a href="../537/algebra.html">Algebra</a><ul class="dropdown-menu"><li><a href="../../../mathsanalysis.html" style="padding-left: 5px; border-bottom: solid 1px #ddd"><i class="fa fa-fw fa-home"></i> Home</a></li><li><a href="../3016/free-access-weekend.html"><i class="fa fa-fw fa-caret-right"></i> Free Access Weekend</a></li><li><a href="../2696/start-here.html"><i class="fa fa-fw fa-caret-right"></i> Start Here</a></li><li><a href="../2902/examination-questions.html"><i class="fa fa-fw fa-caret-right"></i> Examination Questions</a></li><li><a href="../2563/question-bank.html"><i class="fa fa-fw fa-caret-right"></i> Question Bank</a></li><li><a href="../539/functions.html"><i class="fa fa-fw fa-caret-right"></i> Functions</a></li><li><a href="../540/geometry-trigonometry.html"><i class="fa fa-fw fa-caret-right"></i> Geometry & Trigonometry</a></li><li><a href="../549/stats-probability.html"><i class="fa fa-fw fa-caret-right"></i> Stats & Probability</a></li><li><a href="../550/calculus.html"><i class="fa fa-fw fa-caret-right"></i> Calculus</a></li><li><a href="../857/exam-tips.html"><i class="fa fa-fw fa-caret-right"></i> Exam Tips</a></li></ul><div class="pull-right"><a class="sidenav-expand" title="Expand all" href="#"><i class="fa fa-plus-circle"></i></a> <a class="sidenav-compress" title="Compress all" href="#"><i class="fa fa-minus-circle"></i></a></div></h4><ul class="side-nav level-0"><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../662/arithmetic-sequences.html">Arithmetic Sequences</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../661/geometric-sequences.html">Geometric Sequences</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../543/indices-and-logarithms.html">Indices and Logarithms</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../673/counting-principles.html">Counting Principles</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../1732/binomial-theorem-hl.html">Binomial Theorem HL</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../1733/deductive-proofs.html">Deductive Proofs</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../1734/proof-by-contradiction.html">Proof by Contradiction</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../664/proof-by-induction.html">Proof by Induction</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../547/complex-numbers-the-basics.html">Complex Numbers - The Basics</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../674/complex-numbers-de-moivres-theorem.html">Complex Numbers - de Moivre's Theorem</a></label></li><li class="expanded parent selected"><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="complex-numbers-roots-of-polynomials.html">Complex Numbers - Roots of Polynomials</a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> Complex Numbers - Roots of Polynomials <a href="#" class="mark-page-favorite pull-right" data-pid="675" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../537/algebra.html">Algebra</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Complex Numbers - Roots of Polynomials</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/algebra/complex-numbers/roots-of-polynomials/main.png" style="float: left; width: 100px; height: 89px;">The <strong>Conjugate Root</strong> Theorem states that if the complex number <strong><em>a + ib</em></strong> is a root of a polynomial in one variable with real coefficients, then the complex <strong>conjugate</strong> <strong><em>a - bi</em></strong> also a root of that polynomial. This is a useful theorem for solving polynomials with real coefficients. Since the coefficients of the polynomial are real numbers, complex roots must always come in pairs and more than that they must be conjugate pairs - this way two complex numbers can multiply to give a real number.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn to</p> <ul> <li>Understand that complex roots occur in conjugate pairs</li> <li>Solve polynomial equations </li> <li>Make links with this topic and <a href="../681/factor-and-remainder-theorem.html" title="Factor and Remainder Theorem">Factor and Remainder Theorem</a> and <a href="../680/sums-and-products-of-roots.html" title="Sums and Products of Roots">Sums and Products of Roots</a></li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <p>The following video will help you understand all the concepts from this page</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Finding Roots of a Polynomial Equation</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="488"> <p>In the following video we will look at an example involving complex roots of a polynomial equation. Since the coefficients of the polynomial are real numbers, complex roots must always come in pairs and more than that they must be conjugate pairs - this way two complex numbers can multiply to give a real number.</p> <p>The <strong>conjugate root</strong> theorem states that if the complex number <span class="math-tex">\(a+ib\)</span> is a root of a polynomial f(x) in one variable with real coefficients, then the complex <strong>conjugate</strong> <span class="math-tex">\(a-ib\)</span> also a root of that polynomial.</p> <hr> <p>One root of the equation <span class="math-tex">\(4z^4-4z^3-25z^2+55z-42=0\)</span> is <span class="math-tex">\(1+\frac {\sqrt{3}}{2}i\)</span></p> <p>Find the other roots of the equation.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/282630312"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/viete/complex-roots-of-polynomials.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/viete/complex-roots-of-polynomials.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> <div class="smart-object center" data-id="488"> <p>In the following video we will look at an example involving complex roots of a polynomial equation. Since the coefficients of the polynomial are real numbers, complex roots must always come in pairs and more than that they must be conjugate pairs - this way two complex numbers can multiply to give a real number.</p> <p>The <strong>conjugate root</strong> theorem states that if the complex number <span class="math-tex">\(a+ib\)</span> is a root of a polynomial f(x) in one variable with real coefficients, then the complex <strong>conjugate</strong> <span class="math-tex">\(a-ib\)</span> also a root of that polynomial.</p> <hr> <p>One root of the equation <span class="math-tex">\(4z^4-4z^3-25z^2+55z-42=0\)</span> is <span class="math-tex">\(1+\frac {\sqrt{3}}{2}i\)</span></p> <p>Find the other roots of the equation.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/282630312"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/viete/complex-roots-of-polynomials.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/viete/complex-roots-of-polynomials.pdf" width="640"></iframe></p> </section> </div> <div class="smart-object center" data-id="488"> <p>In the following video we will look at an example involving complex roots of a polynomial equation. Since the coefficients of the polynomial are real numbers, complex roots must always come in pairs and more than that they must be conjugate pairs - this way two complex numbers can multiply to give a real number.</p> <p>The <strong>conjugate root</strong> theorem states that if the complex number <span class="math-tex">\(a+ib\)</span> is a root of a polynomial f(x) in one variable with real coefficients, then the complex <strong>conjugate</strong> <span class="math-tex">\(a-ib\)</span> also a root of that polynomial.</p> <hr> <p>One root of the equation <span class="math-tex">\(4z^4-4z^3-25z^2+55z-42=0\)</span> is <span class="math-tex">\(1+\frac {\sqrt{3}}{2}i\)</span></p> <p>Find the other roots of the equation.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/282630312"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/viete/complex-roots-of-polynomials.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/viete/complex-roots-of-polynomials.pdf" width="640"></iframe></p> </section> </div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/roots-of-polynomials/revision-notes_roots_of_polynomials.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/complex-numbers/roots-of-polynomials/revision-notes_roots_of_polynomials.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#5deef021"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="5deef021"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Complex Numbers - Roots of Polynomials <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-449-675" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><em><strong>2 + 3i </strong></em>is a root of a polynomial.</p><p><strong><em>a + bi</em></strong> is another root.</p><p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p></div><div class="q-explanation"><p>According to the conjugate root theorem, complex roots of polynomials come in conjugate pairs.</p><p>If <strong><em>a + bi </em></strong>is a root, then <strong><em>a - bi </em></strong> is a root</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\frac{1}{1-2i} \)</span> is a root of a cubic equation. Which of the following is another root?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac{1-2i}{5}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1+2i}{5}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{-1-2i}{3}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{-1+2i}{3}\)</span></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{1}{1-2i} =\frac{1+2i}{(1-2i)(1+2i)} \)</span></p><p><span class="math-tex">\(\frac{1+2i}{1-4i²} =\frac{1+2i}{5} \)</span></p><p>Another root is the complex conjugate of this = <span class="math-tex">\(\frac{1-2i}{5}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If (z - 2 - i)(z - 2 + i) <span class="math-tex">\(\equiv\)</span> z² + <strong><em>a</em></strong>z + <strong><em>b</em></strong></p><p>work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>(z - 2 - i)(z - 2 + i)</p><p>= z² - 2z + zi</p><p>- 2z + 4 - 2i</p><p>- zi + 2i - i²</p><p>= z² - 4z + 5</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If (z - 1 - 3i)(z - 1 + 3i) <span class="math-tex">\(\equiv\)</span> z² + <strong><em>a</em></strong>z + <strong><em>b</em></strong></p><p>work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p></div><div class="q-explanation"><p>(z - 1 - 3i)(z - 1 + 3i)</p><p>= z² - 2z + 3zi</p><p>- z + 1 - 3i</p><p>- 3zi + 3i - 9i²</p><p>= z² - 2z + 10</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If (2z - 3)(z² - 4z + 8) <span class="math-tex">\(\equiv\)</span> 2z<sup>3 </sup>+ az² + bz - 24</p><p>find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-11"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="28"> <span class="review"></span></p></div><div class="q-explanation"><p>2z(z² - 4z + 8) - 3(z² - 4z + 8)</p><p>= 2z<sup>3</sup> - 8z² + 16z - 3z² + 12z - 24</p><p>= 2z<sup>3</sup> <strong>- 11</strong>z² + <strong>28</strong>z - 24</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If (z - <em><strong>p</strong></em>)(z² + 2z - 1) <span class="math-tex">\(\equiv\)</span> z<sup>3 </sup>- 2z² + az + b</p><p>Work out <strong><em>p</em></strong></p></div><div class="q-answer"><p><strong><em>p</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>Consider the z² term in the expansion of (z - <em><strong>p</strong></em>)(z² + 2z - 1)</p><p>= <span class="math-tex">\(z\times2z-p\times z^2=(2-p)z^2\)</span></p><hr class="hidden-separator"><p>z² term in z<sup>3 </sup>- 2z² + az + b is equal to -2z²</p><p>2 - p = -2</p><p>p = 4</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If (z - <em><strong>p</strong></em>)(z² - 4z + 5) <span class="math-tex">\(\equiv\)</span> z<sup>3 </sup> + az² - 3z + b</p><p>Work out <strong><em>p</em></strong></p></div><div class="q-answer"><p><strong><em>p</em></strong> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p>Consider the z term in the expansion of (z - <em><strong>p</strong></em>)(z² - 4z + 5)</p><p><span class="math-tex">\(=z\times5-p\times(-4z)=(5+4p)z\)</span></p><hr class="hidden-separator"><p>z term in z<sup>3 </sup> + az² - 3z + b is equal to -3z</p><p>5 + 4p = - 3</p><p>p = - 2</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>How many roots does the following cubic have?</p><p><img alt="" src="../../files/algebra/complex-numbers/roots-of-polynomials/quiz/q8.bmp" style="width: 300px; height: 215px;"></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> real root(s)</p><p> <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> complex root(s)</p></div><div class="q-explanation"><p>The graph cuts the x axis 3 times.</p><p>Therefore, it has 3 real roots.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>How many roots does the following cubic have?</p><p><img alt="" src="../../files/algebra/complex-numbers/roots-of-polynomials/quiz/q9.bmp" style="width: 300px; height: 215px;"></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> real root(s)</p><p> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> complex root(s)</p></div><div class="q-explanation"><p>The graph cuts the x axis 1 time.</p><p>Therefore, it has 1 real root.</p><p>There are 3 roots altogether, so it must have 2 complex roots.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>How many roots does the following quartic have?</p><p><img alt="" src="../../files/algebra/complex-numbers/roots-of-polynomials/quiz/q10.bmp" style="width: 300px; height: 215px;"></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> real root(s)</p><p> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> complex root(s)</p></div><div class="q-explanation"><p>The graph cuts the x axis 2 times.</p><p>Therefore, it has 2 real roots.</p><p>There are 4 roots altogether, so it must have 2 complex roots.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="484"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>One root of the equation <em>z² + bz + c = 0</em> is <em>2+3i</em> where <span class="math-tex">\(b,c\in\mathbb{Z}\)</span>.</p> <p>Find the value of <strong><em>b</em></strong> and the value of <strong><em>c</em></strong>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">If z=2+3i is a root of the equation, then the complex conjugate z=2-3i is also a root of the equation.<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/roots-of-polynomials/esq_complex_roots_polynomials1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/roots-of-polynomials/esq_complex_roots_polynomials1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="842"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><span class="math-tex">\(\frac{2}{1+i}\)</span> is a root to the quadratic equation z² + px + q = 0</p> <div> <p>a) Find the other root</p> </div> <p>b) Hence find the values of <strong><em>p</em></strong> and <strong><em>q</em></strong>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content>Make the denomiantor of the root <span class="math-tex">\(\frac{2}{1+i}\)</span> real by multiplying the numerator and denominator by the complex conjugate and find the other root.</section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/roots-of-polynomials/esq_roots_polynomials1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/roots-of-polynomials/esq_roots_polynomials1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="843"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><strong><em>2 - 3i</em></strong> is a root of the equation <span class="math-tex">\(z^3-7z^2+az+b=0\quad ,a, b\in\ \mathbb{R}\)</span></p> <p>Work out <strong><em>a</em></strong> and <strong><em>b </em></strong>and the other roots of the equation.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><strong><em>2 + 3i</em></strong> is also a root to the equation</p> <p><content>Multiply the two factors <strong><em>(z - 2 + 3i)</em></strong> and <strong><em>(z - 2 - 3i)</em></strong></content></p> <p>and then find the linear real factor</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/roots-of-polynomials/esq_roots_polynomials3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/roots-of-polynomials/esq_roots_polynomials3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="844"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The quartic equation <span class="math-tex">\(z^4+az^3+bz^2+cz+d\)</span> has roots <em><strong>2 + i </strong></em>and <strong><em>2i</em></strong></p> <div> <p>a) Work out the other roots of the equation</p> </div> <p>b) Find the values of a , b , c and d</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>If <strong><em>2i </em></strong>is a root, then <strong><em>-2i</em></strong> is one too.</p> <p><content>You can multiply the factors <strong><em>(z - 2i)</em></strong> and <strong><em>(z + 2i)</em></strong> together to find a quadratic.</content></p> <p>Repeat this for the other root <strong><em>2 + i</em></strong></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/roots-of-polynomials/esq_roots_polynomials4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/roots-of-polynomials/esq_roots_polynomials4.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="485"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The equation <span class="math-tex">\(2z^{ 4 }−9z^{ 3 }+pz^{ 2 }+qz−174=0 \quad,\quad p,q\in\mathbb{Z}\)</span> has two real roots <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span> and two complex roots <span class="math-tex">\(\gamma\)</span> and <span class="math-tex">\(\delta\)</span> where <span class="math-tex">\(\gamma=2-5i\)</span>.</p> <p>a. Show that <span class="math-tex">\(\alpha+\beta=\frac{1}{2}.\)</span></p> <p>b. Find <span class="math-tex">\(\alpha\beta\)</span>.</p> <p>c. <strong>Hence</strong> find the two real roots α and β.</p> <p>d. Find the values of <strong><em>p</em></strong> and <strong><em>q</em></strong>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>a. If <em>γ=2−5i</em> is a root…then <em>δ = 2+5i</em> is also a root. Work out γ+δ and sum of 4 roots</p> <p>b. Work out γδ and the product of the 4 roots</p> <p>c. Work out α and β using the two equations for <span class="math-tex">\(\alpha+\beta\)</span> and <span class="math-tex">\(\alpha\beta\)</span>.</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/roots-of-polynomials/esq_complex_roots_polynomials2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/complex-numbers/roots-of-polynomials/esq_complex_roots_polynomials2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="675"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Complex Numbers - Roots of Polynomials</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer text-xs-center"> <span id="last-edited" class="mb-xs-3"> </span> <div class="actions mt-xs-3"> <button id="save-my-progress" type="button" class="btn btn-sm btn-primary text-center btn-xs-block"> <i class="fa fa-fw fa-floppy-o"></i> Save </button> </div> </div></div> <div id="modal-feedback" class="modal fade" tabindex="-1" role="dialog"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <h4 class="modal-title">Feedback</h4> <button type="button" class="close hidden-xs hidden-sm" data-dismiss="modal" aria-label="Close"> <span aria-hidden="true">×</span> </button> </div> <div class="modal-body"> <div class="errors"></div> <p><strong>Which of the following best describes your feedback?</strong></p> <form method="post" style="overflow: hidden"> <div class="form-group"> <div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Recommendation"> Recommend</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Problem"> Report a problem</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Improvement"> Suggest an improvement</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Other"> Other</label></div> </div> <hr> <div class="row"> <div class="col-md-6"> <div class="form-group"> <label for="feedback-name">Name</label> <input type="text" class="form-control" name="feedback-name" placeholder="Name" value=" "> </div> </div> <div class="col-md-6"> <div class="form-group"> <label for="feedback-email">Email address</label> <input type="email" class="form-control" name="feedback-email" placeholder="Email" value="@airmail.cc"> </div> </div> </div> <div class="form-group"> <label for="feedback-comments">Comments</label> <textarea class="form-control" name="feedback-comments" style="resize: vertical;"></textarea> </div> <input type="hidden" name="feedback-ticket" value="082b9c9c4ae3624d"> <input type="hidden" name="feedback-url" value="https://studyib.net/mathsanalysis/page/675/complex-numbers-roots-of-polynomials"> <input type="hidden" name="feedback-subject" value="11"> <input type="hidden" name="feedback-subject-name" value="Maths: Analysis & Approaches"> <div class="pull-left"> </div> </form> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary btn-xs-block feedback-submit mb-xs-3 pull-right"> <i class="fa fa-send"></i> Send </button> <button type="button" class="btn btn-default btn-xs-block m-xs-0 pull-left" data-dismiss="modal"> Close </button> </div> </div> </div></div> </article> <hr class="hidden-md hidden-lg"> <div class="hidden-md hidden-lg mt-xs-3"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <input type="hidden" id="user-id" value="38342"></div><input id="ticket" type="hidden" value="082b9c9c4ae3624d"><input id="tzoffset" type="hidden" value="new"><input id="fp" class="fp" type="hidden" value=""></div><div id="std-footer"> <div class="wmap"> <div class="layout-wrapper"> <p> <a href="https://www.inthinking.net"> © <span id="footer-year"></span> <em>InThinking</em> <script>document.getElementById("footer-year").innerHTML = new Date().getFullYear();</script> </a> | <a target="_self" href="../../../about.html"> About us </a> | <a target="_self" href="../../../terms-and-conditions.html"> Legal </a> | <a target="_self" href="../../../contact.html"> Contact </a> </p> <p> <a class="social" target="_blank" href="https://twitter.com/#!/inthinker"> <img src="../../../img/social/twitter-square.svg"> Twitter </a> <a class="social" target="_blank" href="https://www.facebook.com/inthinking.net"> <img src="../../../img/social/facebook-square.svg"> Facebook </a> <a class="social" target="_blank" href="https://www.linkedin.com/profile/view?id=139741989"> <img src="../../../img/social/linkedin-square.svg"> LinkedIn </a> </p> </div> </div></div><script src="../../../js/jquery-1.11.3.min.js"></script><script src="../../../js/bootstrap.min.js"></script><script src="../../../js/jquery-fancybox/jquery.fancybox.min.js"></script><script>var pageId = 675;</script><script type="text/javascript" src="../../../js/jqueryui/jquery-ui-custom.min.js"></script><script type="text/javascript" src="../../../js/jqueryui/jquery.ui.touch-punch.js"></script><script type="text/javascript" src="../../../js/std-quizzes-helpers.min.js?v=20220406"></script><script src="../../../ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script><script type="text/javascript" src="../../../js/slick-carousel/slick.min.js"></script><script type="text/javascript" src="../../../js/std-slide-quizzes.min.js?v=20220406"></script><script type="text/javascript" src="../../../js/jqueryui/jquery-ui.min.js"></script><script type="text/javascript" src="../../../js/jqueryui/jquery.ui.touch-punch.min.js"></script><script src="../../../js/user/page-my-progress.min.js?v=202211221000"></script><script>var sAJAX='/pages/subjects/activity/user-stats-page.php?t=082b9c9c4ae3624d&p=675&s=11&x=14488';var sData='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';var loopSecs = 30;var minSecsLog = 60;var lsKey = '11-83608-675';</script><script src="../../../js/subjects/activity/user-stats-page.min.js?v=202205311700"></script><script src="../../../js/subjects/activity/my-favorites.min.js?v=20220610"></script><script src="../../../js/subjects/feedback.min.js?v=202211221000"></script><script>var sessionUpdateSecs = 300;</script><script type="text/javascript" src="../../../js/session-updater.js"></script><script>(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)})(window,document,'script','https://www.google-analytics.com/analytics.js','ga');ga('create', 'UA-98480796-4', 'auto');ga('send', 'pageview');</script><script src="../../../js/devicefp/devicefp.min.js"></script><script src="../../../js/jq-quickfit/jq-quickfit.min.js?v=20220201"></script><script src="../../../js/frontpage-subjects.min.js?v=202211161300"></script><script type="text/javascript" src="../../../js/studyib-jq.min.js?v=202211241300"></script><script type="text/javascript">$(document).ready(function(){ $('#trigger-modal-video-overview').click(function() { if( $('#modal-video-overview .modal-body iframe').attr('src').length == 0 ) { $('#modal-video-overview .modal-body iframe').attr('src', 'https://www.youtube.com/embed/hxxtdiLdTFk?rel=0'); } }); $('#modal-video-overview').on('hidden.bs.modal', function() { $('#modal-video-overview .modal-body iframe').attr('src', ''); }); });</script></body></html>