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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="../539/functions.html">Functions</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Sums and Products of Roots</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/functions/viete/main.jpg" style="float: left; width: 100px; height: 100px;"></p> <p>On this page, we learn all about roots of polynomial equations and how to use the formulae for the sum of the roots and the product of the roots. For the polynomial equation, degree <strong><em>n</em></strong></p> <p><span class="math-tex">\(\sum _{ r=1 }^{ n }{ { a }_{ r } } { x }^{ r }\)</span> , the sum of the roots = <span class="math-tex">\(\frac { { a }_{ n-1 } }{ { a }_{ n } } \)</span> , the product of the roots =<span class="math-tex">\((-1)^n\frac { { a }_{ 0 } }{ { a }_{ n } } \)</span></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 about</p> <ul> <li>roots of polynomial equations</li> <li>sums and products of roots</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"> <p>The following videos 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>Where do the formulae come from</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="489"> <p>Below you will find the formula for finding the sum of the roots and the product of the roots of polynomial equations of different degrees, but what are they and where do these formulae come from? We will find out in the video below.</p> <p><img alt="" src="../../files/functions/viete/summarytable2.jpg" style="width: 100%;"></p> <p>* These formulae do not appear in the formula booklet. You should learn the general case for degree <em>n</em> by heart.</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/282495541"></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/sum-and-product-of-roots-explanation.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/viete/sum-and-product-of-roots-explanation.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow 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>Finding New Polynomials</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="486"> <p>In the following video, we look at a typical exam-style question in which we are required to find a new polynomial having worked out the sum and product of roots of this equation:</p> <hr> <p>The quadratic equation x² - 4x + 5 = 0 has roots <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span>.</p> <p>a. Without solving the equation, find the value of</p> <p style="margin-left: 40px;">i <span class="math-tex">\(\alpha + \beta\)</span></p> <p style="margin-left: 40px;">ii <span class="math-tex">\(\alpha \beta\)</span></p> <p>b. Another quadratic equation 5x² + bx + c = 0 , b,c<span class="math-tex">\(\in \mathbb{Z}\)</span> has roots <span class="math-tex">\(\frac{1}{\alpha}\)</span> and <span class="math-tex">\(\frac{1}{\beta}\)</span>.</p> <p>Find the value of <strong><em>b </em></strong> and the value of <strong><em>c</em></strong>.</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/282505525"></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/finding-new-polynomial-equations.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/viete/finding-new-polynomial-equations.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow 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>Problem Solving</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="487"> <p>The following video is a typical exam-style question. One of the things that IB likes to do in examinations, is to ask a question that requires knowledge from different areas of the course. This can be quite challenging and often something that textbooks do not give you practice in. This question is about sums and products of roots of polynomial equations, but requires some interesting problem-solving skils, since the roots of the equation are terms of an arithmetic sequence. Here is the question:</p> <hr> <p>Consider the equation <span class="math-tex">\(64x^{ 3 }−144x^{ 2 }+92x−15=0\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>Write down the numerical value of the sum and the product of the roots of this equation.</li> <li>The roots of this equation are three consecutive terms of an arithmetic sequence. Solve the equation.</li> </ol> <p style="text-align: center"><iframe allow="autoplay; fullscreen" allowfullscreen="" frameborder="0" height="480" src="https://player.vimeo.com/video/431586578" width="640"><br> </iframe></p> <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/sum-and-product-of-roots---problem-solving.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/viete/sum-and-product-of-roots---problem-solving.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow 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>Complex Roots</p> </div> </div> <div class="panel-body"> <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 class="panel-footer"> <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/functions/viete/revision-notes_sum_and_product_of_roots.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/functions/viete/revision-notes_sum_and_product_of_roots.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 sums and products of <strong>real roots </strong>of polynomial</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#f21b7fc7"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="f21b7fc7"> <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%;"> Sums and Products of Roots <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="11-207-680" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Consider the equation 2x²-8x+5=0</p><p>Find the sum and the product of the roots.</p></div><div class="q-answer"><p>sum of roots = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p>product of roots = <input type="text" style="height: auto;" data-c="2.5"> <span class="review"></span></p></div><div class="q-explanation"><p>For the quadratic equation <span class="math-tex">\(a_2x^2+a_1x+a_0=0\)</span></p><p>sum of roots = <span class="math-tex">\(\frac {-a_1}{a_2}\)</span></p><p>product of roots = <span class="math-tex">\(\frac {a_0}{a_2}\)</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><span class="math-tex">\(\alpha ,\beta \ and\ \gamma\)</span> are roots of the polynomial equation <span class="math-tex">\(x^3-6x^2+11x-6=0\)</span></p><p>Find <span class="math-tex">\(\alpha +\beta + \gamma\)</span> and <span class="math-tex">\(\alpha\beta \gamma\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\alpha +\beta + \gamma\)</span> = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p><p><span class="math-tex">\(\alpha\beta \gamma\)</span> = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>For the cubic equation <span class="math-tex">\(a_3x^3+a_2x^2+a_1x+a_0=0\)</span></p><p>sum of roots = <span class="math-tex">\(\frac {-a_2}{a_3}\)</span></p><p>product of roots = <span class="math-tex">\(\frac {-a_0}{a_3}\)</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>Consider the equation <span class="math-tex">\(4x^4-10x^3-5x^2+3x=6\)</span></p><p>Find the sum and the product of the roots.</p></div><div class="q-answer"><p>sum of roots = <input type="text" style="height: auto;" data-c="2.5"> <span class="review"></span></p><p>product of roots = <input type="text" style="height: auto;" data-c="-1.5"> <span class="review"></span></p></div><div class="q-explanation"><p>For the quartic equation <span class="math-tex">\(a_4x^4+a_3x^3+a_2x^2+a_1x+a_0=0\)</span></p><p>sum of roots = <span class="math-tex">\(\frac {-a_3}{a_4}\)</span></p><p>product of roots = <span class="math-tex">\(\frac {a_0}{a_4}\)</span></p><p>Note that the equation rearranges to <span class="math-tex">\(4x^4-10x^3-5x^2+3x-6=0\)</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>The product of the roots of the quadratic equation px² - 3x - 2 = 0 is equal to 1.</p><p>Find <em><strong>p</strong></em>.</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>For the quadratic equation <span class="math-tex">\(a_2x^2+a_1x+a_0=0\)</span></p><p>product of roots = <span class="math-tex">\(\frac {a_0}{a_2}\)</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>The polynomial equation <span class="math-tex">\(6x^3+px^2-11x=q\)</span> has roots <span class="math-tex">\(\alpha ,\beta \ and\ \gamma\)</span>.</p><p> <span class="math-tex">\(\alpha +\beta + \gamma=\frac{1}{6}\)</span> </p><p><span class="math-tex">\(\alpha\beta \gamma=-1\)</span></p><p>Find <strong><em>p </em></strong> and <strong><em>q</em></strong>.</p></div><div class="q-answer"><p><strong><em>p</em></strong> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p><p><strong><em>q</em></strong> = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span></p></div><div class="q-explanation"><p>The polynomial equation becomes <span class="math-tex">\(6x^3+px^2-11x-q=0\)</span></p><p><span class="math-tex">\(\alpha +\beta + \gamma=\frac{1}{6}=-\frac{p}{6}\)</span></p><p><span class="math-tex">\(\alpha\beta \gamma=-1=-\frac{-q}{6}\\ -1=\frac{q}{6}\)</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>Which of the following is equal to <span class="math-tex">\( {\alpha }^{ 2}+{\beta }^{ 2 } \)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\({ \left( \alpha +\beta \right) }^{ 2 }-2\alpha \beta\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({ \left( \alpha +\beta \right) }^{ 2 }+2\alpha \beta\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({ \left( \alpha +\beta \right) }^{ 2 }-\alpha \beta\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({ \left( \alpha +\beta \right) }^{ 2 }+\alpha \beta\)</span></label></p></div><div class="q-explanation"><p><label class="radio"><span class="math-tex">\({ \left( \alpha +\beta \right) }^{ 2 }-2\alpha \beta\\ { \left( \alpha +\beta \right) }{ \left( \alpha +\beta \right) }-2\alpha \beta\\ =\alpha^2+\alpha\beta +\alpha\beta+\beta^2-2\alpha \beta\\ =\alpha^2+\beta^2\)</span></label></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>The roots of a polynomial equation are <span class="math-tex">\(\alpha \)</span> and <span class="math-tex">\(\beta \)</span></p><p><span class="math-tex">\(\alpha +\beta =6\)</span></p><p><span class="math-tex">\(\alpha\beta =-2\)</span></p><p>Find <span class="math-tex">\(\frac{1}{\alpha}+\frac{1}{\beta}\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\frac{1}{\alpha}+\frac{1}{\beta}\)</span> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\beta}{\alpha\beta}+\frac{\alpha}{\alpha\beta}\\ =\frac{\alpha+\beta}{\alpha\beta}=\frac{6}{-2}\)</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>The roots of a quadratic equation are <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span> , where <span class="math-tex">\(\alpha<\beta\)</span>.</p><p>The sum of the roots is equal to -1</p><p>The product of the roots is equal to -6.</p><p>Find the roots.</p></div><div class="q-answer"><p><span class="math-tex">\(\alpha\)</span> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p><p><span class="math-tex">\(\beta\)</span> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>Solve the equations:</p><p><span class="math-tex">\(\alpha+\beta=-1\)</span></p><p><span class="math-tex">\(\alpha\beta =- 6\)</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>A polynomial equation f(x)=0, has 3 roots</p><p>The sum of the roots is equal to 10.</p><p>A new polynomial is defined by g(x)=f(x+2)</p><p>Find the sum of the roots of the equation g(x)=0</p></div><div class="q-answer"><p>sum of roots = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation">g(x)=f(x+3) represents a shift of 2 units to the left.</div><div class="q-explanation">Hence, each root is 2 units less.</div><div class="q-explanation">Sum of roots = 10 - 3x2</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">In the quadratic equation 4x² + px - 15 = 0 , one root is 4 more than the other. Find <strong><em>p </em></strong>given that <strong><em>p</em></strong>>0</div><div class="q-answer"><p><em><strong>p</strong></em> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>Let the roots be <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\alpha+4\)</span></p><p>Product of roots =<span class="math-tex">\(\alpha(\alpha+4)=-\frac{15}{4}\)</span></p><p><span class="math-tex">\(4\alpha(\alpha+4)=-15\\ 4\alpha^2+16\alpha+15=0\\ (2\alpha+3)(2\alpha+5)=0\\ \alpha=-\frac{3}{2} \quad, \quad \alpha=-\frac{5}{2}\)</span></p><hr><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p>If <span class="math-tex">\( \alpha=-\frac{5}{2} \quad \alpha+4=\frac{3}{2}\)</span></p><p>a(2x+5)(2x-3)=0 , a must be 1</p><p>4x²+4x-15=0</p></td><td><p>If <span class="math-tex">\( \alpha=-\frac{3}{2} \quad \alpha+4=\frac{5}{2}\)</span></p><p>a(2x+3)(2x-5)=0 , a must be 1</p><p>4x²-4x-15=0</p></td></tr></tbody></table><p>p>0</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> </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="480"> <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>The quadratic equation <span class="math-tex">\(3x^{ 2 }−8x+2=0\)</span> has roots <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span>.</p> <p>a. Without solving the equation, find the value of <span class="math-tex">\(\alpha + \beta\)</span> and <span class="math-tex">\(\alpha \beta\)</span>.</p> <p>b. Another quadratic equation <span class="math-tex">\(3x^{ 2 }+bx+c=0\quad ,\quad b,c\in \mathbb{Z}\)</span> has roots <span class="math-tex">\(\frac{\alpha}{\beta}\)</span> and <span class="math-tex">\(\frac{\beta}{\alpha}\)</span>. Find the value of <em><strong>b</strong></em> and <em><strong>c</strong></em>.</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>b. Write the sum and product of <span class="math-tex">\(\frac{\alpha}{\beta}\)</span> and <span class="math-tex">\(\frac{\beta}{\alpha}\)</span> in terms of <span class="math-tex">\(\alpha + \beta\)</span> and <span class="math-tex">\(\alpha \beta\)</span>.</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/functions/viete/esq_viete1a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/viete/esq_viete1a.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="481"> <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>In the quadratic equation <span class="math-tex">\(px^{ 2 }−45x+25=0\quad ,\quad p\in \mathbb{Z}\)</span> ,one root is two times the other.</p> <p>Find the value of <em><strong>p</strong></em>.</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">Let the roots be <span class="math-tex">\(\alpha, 2\alpha\)</span><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/functions/viete/esq_viete2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/viete/esq_viete2.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="482"> <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>Let <span class="math-tex">\(f(x)=2x^{ 4 }+x^{ 3 }−14x^{ 2 }+5x+6\quad ,\quad x\in \mathbb{R}\)</span></p> <p>a. For the polynomial equation f(x) = 0 , find the value of</p> <ol> <li>the sum of the roots</li> <li>the product of the roots</li> </ol> <p>b. A new polynomial is defined by g(x) = f(x - 2). Find the sum of the roots of the equation g(x) = 0</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">b. What transformation is represented by g(x) = f(x - 2)? What would happen to the position of the roots?<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/functions/viete/esq_viete3a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/viete/esq_viete3a.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="483"> <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>Consider the equation <span class="math-tex">\(8x^{ 3 }−42x^{ 2 }+px−27=0.\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>State <ol> <li>the sum of the roots of the equation</li> <li>the product of the roots of the equation</li> </ol> </li> <li>The roots of this equation are three consecutive terms of a geometric sequence. Taking the roots to be <span class="math-tex">\(\frac{\alpha}{\beta},\alpha,\alpha\beta\)</span> show that one of the roots is <span class="math-tex">\(\frac{3}{2}\)</span></li> <li>Solve the equation.</li> <li>Find the value of <strong><em>p</em></strong>.</li> </ol> <p><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>b. What is the product of the three roots <span class="math-tex">\(\frac{\alpha}{\beta},\alpha,\alpha\beta\)</span>?</p> <p>c. What is the sum of the three roots <span class="math-tex">\(\frac{\alpha}{\beta},\alpha,\alpha\beta\)</span>?</p> <p>d. The equation is <span class="math-tex">\(a(x-\frac{\alpha}{\beta})(x-\alpha)(x-\alpha\beta)=0\)</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/functions/viete/esq_viete4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/viete/esq_viete4.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 5</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 6</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 class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="680"> 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