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</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">Factor and Remainder Theorem</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/factor-and-remainder-theorem/main.jpg" style="float: left; width: 100px; height: 100px;"> The Factor and Remainder Theorem form a small, but significant part of the course. It is not at all difficult, but it is worth reminding yourself of these theorem, since, in examinations students often forget them and waste a lot of time using long methods instead. The Factor Theorem is a theorem that allows us to find factors of polynomial functions, to find zeros and ultimately to help us sketch graphs.</p> <p>A polynomial function f(x) is an algebraic expression that takes the form</p> <p style="margin-left: 40px;"><span class="math-tex">\({ f(x)=a }_{ n }{ x }^{ n }+{ a }_{ n-1 }{ x }^{ n-1 }+{ a }_{ n-2 }{ x }^{ n-2 }+\quad ...\quad +{ a }_{ 1 }{ x }^{ 1 }+{ a }_{ 0 }\)</span></p> <p>For example, a polynomial function of degree 3 is a cubic function e.g <span class="math-tex">\(f(x) = 7x^3+4x^2-3x+4\)</span></p> <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>polynomial functions</li> <li>zeros, roots and factors</li> <li>the factor and remainder theorem</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>Factorising Polynomials</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="477"> <p>The <strong>Remainder Theorem</strong> states that for a polynomial f(x),</p> <p style="margin-left: 80px;"><strong>the remainder when divided by (x-a) is f(a).</strong></p> <p>Example</p> <p>Let <span class="math-tex">\(f(x)=2x^3+4x^2+x-5\)</span></p> <p>If we divide this polynomial by <span class="math-tex">\((x-2)\)</span>...</p> <p><span class="math-tex">\(\frac{2x^3+4x^2+x-5}{x-2}=2x^2+8x+17+\frac{29}{x-2}\)</span></p> <p>... we get a quotient <span class="math-tex">\(2x^2+8x+17\)</span> and a <strong>remainder 29</strong></p> <p>When we work out f(2)...</p> <p><span class="math-tex">\(f(2)=2(2)^3+4(2)^2+(2)-5=29\)</span>...</p> <p>... we get <strong>29</strong></p> <hr> <p>The factor theorem states that for a polynomial f(x),</p> <p style="margin-left: 80px;"><strong>(x-a) is a factor if and only if f(a)=0</strong></p> <p>Example</p> <p>Let <span class="math-tex">\(f(x)=x^4+2x^3-7x^2-8x+12\)</span></p> <p>When we work out f(1)...</p> <p><span class="math-tex">\(f(1)=(1)^4+2(1)^3-7(1)^2-8(1)+12 = 0\)</span></p> <p>If we divide this polynomial by <span class="math-tex">\((x-1)\)</span>...</p> <p><span class="math-tex">\(\frac{x^4+2x^3-7x^2-8x+12}{x-1}=x^3+3x^2-4x-12\)</span></p> <p>... there is no remainder, <span class="math-tex">\((x-1)\)</span> is a factor of the polynomial</p> <hr> <p>In this video, we will look at an application of the factor theorem which is to factorize polynomials:</p> <p><strong>Factorize completely <span class="math-tex">\(f(x)=2x^3+x^2-7x-6\)</span></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/281949892"></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/factor-and-remainder-theorem/factor-and-remainder-theorem.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/factor-and-remainder-theorem/factor-and-remainder-theorem.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"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Typical Exam-style Question</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="478"> <p lang="fr" style="margin:0in;font-family:Calibri;font-size:14.0pt">In the following video, we will look at a typical exam-style question:</p> <p lang="fr" style="margin:0in;font-family:Calibri;font-size:14.0pt">The cubic polynomial <span class="math-tex">\(f(x)=ax^3+bx^2-29x+60\)</span> has a factor <span class="math-tex">\((x+4)\)</span> and leaves a remainder of 6 when divided by <span class="math-tex">\((x-2)\)</span>.</p> <p lang="fr" style="margin:0in;font-family:Calibri;font-size:14.0pt">a. Find the value of <strong><em>a</em></strong> and <strong><em>b</em></strong>.</p> <p lang="fr" style="margin:0in;font-family:Calibri;font-size:14.0pt">b. Factorize the polynomial</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/281969860"></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/factor-and-remainder-theorem/factor-and-remainder-theorem-exam-style-question.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/factor-and-remainder-theorem/factor-and-remainder-theorem-exam-style-question.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="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/factor-and-remainder-theorem/revision-notes_factor_and_remainder_theorem.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/functions/factor-and-remainder-theorem/revision-notes_factor_and_remainder_theorem.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="#3d1e5616"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="3d1e5616"> <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%;"> Factor and Remainder Theorem <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="11-206-681" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Given that f(x) = x<sup>3</sup>+6x<sup>2</sup>+14x+15 , find f(1)</p></div><div class="q-answer"><p>f(1) = <input type="text" style="height: auto;" data-c="36"> <span class="review"></span> </p></div><div class="q-explanation"><p>f(1) = 1<sup>3</sup>+6(1)<sup>2</sup>+14(1)+15 = 36</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>Given that f(x) = x<sup>3</sup>+6x<sup>2</sup>+14x+15 , find f(-3)</p></div><div class="q-answer"><p>f(-3) = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p>f(-3) = (-3)<sup>3</sup>+6(-3)<sup>2</sup>+14(-3)+15 = -27 + 54 - 42 + 15 = 0</p><p>note that x - 3 would be a factor of this polynomial</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>Find the remainder when x<sup>3</sup> + 2x<sup>2</sup> - 5x - 6 is divided by x - 1</p></div><div class="q-answer"><p>Remainder = <input type="text" style="height: auto;" data-c="-8"> <span class="review"></span></p></div><div class="q-explanation"><p>Let f(x) = x<sup>3</sup> +2x<sup>2</sup> -5x - 6</p><p>Remainder = f(1) = (1)<sup>3</sup> +2(1)<sup>2</sup> - 5(1) - 6 = 1 + 2 - 5 - 6 = -8</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>Find the remainder when x<sup>4</sup> - 3x<sup>3</sup> - 7x<sup>2</sup> + 15x + 18 is divided by x + 2</p></div><div class="q-answer"><p>Remainder = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p>Remainder = f(-2) = (-2)<sup>4</sup> - 3(-2)<sup>3</sup> - 7(-2)<sup>2</sup> + 15(-2) + 18 = 16 + 24 - 28 - 30 +18 = 0</p><p>Hence x + 2 is a factor.</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>f(x) is a polynomial of degree 3</p><p>f(1) = 3</p><p>f(2) = -4</p><p>f(1.5) = 0</p><p>Which of the following statements is definitely true?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 2x - 3 is a factor of f(x)</label></p><p><label class="radio"><input type="radio"> 2x + 3 is a factor of f(x)</label></p><p><label class="radio"><input type="radio"> 3x - 2 is a factor of f(x)</label></p><p><label class="radio"><input type="radio"> 3x + 2 is a factor of f(x)</label></p></div><div class="q-explanation"><p>If <span class="math-tex">\(f(\frac {3}{2})=0\)</span> then <span class="math-tex">\((x - \frac{3}{2})\)</span> is a factor or (2x - 3)</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>f(x) is a polynomial of degree 3 and f(-1) = f(1) = f(2) = 0.</p><p>Which of the following could be the function f(x)?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> f(x) = (x² - 1)(x - 2)</label></p><p><label class="radio"><input type="radio"> f(x) = (x + 1)(x - 1)(x + 2)</label></p><p><label class="radio"><input type="radio"> f(x) = (2x - 1)(x + 1)(x - 1)</label></p><p><label class="radio"><input type="radio"> f(x) = (2x + 1)(x - 1)(x + 1)<span class="radio"></span></label></p></div><div class="q-explanation"><p>f(-1) = 0 implies x + 1 is a factor</p><p>f(1) = 0 implies x - 1 is a factor</p><p>f(2) = 0 implies x - 2 is a factor</p><p>f(x) = (x + 1)(x - 1)(x - 2)<label class="radio"></label></p><p>f(x) =(x² - 1)(x - 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>f(x) is a polynomial of <strong>degree</strong> <strong>4</strong></p><p>f(-1) = f(2) = f(3) = 0</p><p>Which of the following could be a graph of <strong><em>f</em></strong></p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p style="text-align: center;"><img alt="" src="../../files/functions/factor-and-remainder-theorem/quiz1" style="width: 300px; height: 238px;"></p><p style="text-align: center;"><strong>A</strong></p></td><td><p style="text-align: center;"><img alt="" src="../../files/functions/factor-and-remainder-theorem/quiz2" style="width: 300px; height: 243px;"></p><p style="text-align: center;"><strong>B</strong></p></td></tr><tr><td><p style="text-align: center;"><img alt="" src="../../files/functions/factor-and-remainder-theorem/quiz3" style="width: 300px; height: 262px;"></p><p style="text-align: center;"><strong>C</strong></p></td><td><p style="text-align: center;"><img alt="" src="../../files/functions/factor-and-remainder-theorem/quiz4" style="width: 300px; height: 268px;"></p><p style="text-align: center;"><strong>D</strong></p></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> C</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> D</label></p></div><div class="q-explanation"><p>The key piece of information is that the polynomial is of degree <strong>4</strong></p><p><strong>Hence it has 4 roots with one repeated</strong></p><p><strong><img alt="" src="../../files/functions/factor-and-remainder-theorem/quiz5.jpg" style="width: 539px; height: 530px;"></strong></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 f(x) = 3x<sup>3</sup> + ax<sup>2</sup> + x - 10 is divisible by (x - 2). Find the value of <strong><em>a</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></div><div class="q-explanation"><p>If f(x) is divisible by (x - 2) , then (x - 2) is a factor.</p><p>f(2) = 3(2)<sup>3</sup> + a(2)<sup>2</sup> + (2) - 10 = 0</p><p>24 + 4a + 2 - 10 = 0</p><p>4a = -16</p><p>a = -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>When the polynomial f(x) = x<sup>4</sup> + 4x<sup>3</sup> - 2x<sup>2</sup> - 12x + <strong><em>a</em></strong> is divided by (x + 1) it leaves a remainder of 16.</p><p>Find <strong><em>a</em></strong>.</p></div><div class="q-answer"><p><strong><em>a </em></strong>= <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p></div><div class="q-explanation"><p>f(-1) = (-1)<sup>4</sup> + 4(-1)<sup>3</sup> - 2(-1)<sup>2</sup> - 12(-1) + <strong><em>a </em></strong>= 16</p><p>1 - 4 - 2 + 12 + <strong><em>a</em></strong> = 16</p><p><strong><em>a</em></strong> +7 = 16</p><p><strong><em>a</em></strong> = 9</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>Complete the gaps to make the following statements true</p></div><div class="q-answer"><p>Put rounded brackets around your factor</p><p>3x<sup>3</sup> - x<sup>2</sup> - 20x - 12 = (x² - x - 6)<input type="text" style="height: auto;" data-c="(3x+2)"> <span class="review"></span></p><p>2x<sup>3</sup> + 5x<sup>2</sup> - 2x - 5 = (x² - 1)<input type="text" style="height: auto;" data-c="(2x+5)"> <span class="review"></span></p><p>4x<sup>4</sup> - 15x<sup>3</sup> - 7x<sup>2</sup> + 60x - 36 = x<sup>3</sup> - 3x<sup>2</sup> - 4x + 12<input type="text" style="height: auto;" data-c="(4x-3)"> <span class="review"></span></p></div><div class="q-explanation"></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> <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="476"> <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 cubic polynomial <span class="math-tex">\(3x^3+ax^2+bx-12\)</span> has a factor <span class="math-tex">\((x-2)\)</span> and leaves a remainder of <strong>-20</strong> when divided by <span class="math-tex">\((x-1)\)</span>.</p> <p>Find the value of <strong><em>a</em></strong> and <strong><em>b</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>Use the Factor and Remainder Theorem</p> <p><content>f(2)=0</content></p> <p>f(1)=-20</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/factor-and-remainder-theorem/esq_functions_remainder_th1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/factor-and-remainder-theorem/esq_functions_remainder_th1.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="475"> <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>Given that <span class="math-tex">\(ax^3+bx^2+17x-6\)</span> is exactly divisible by <span class="math-tex">\((x-1)(x-2)\)</span>, find the value of <strong><em>a</em></strong> and <strong><em>b</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>Use the Factor and Remainder Theorem</p> <p><content>f(1)=0</content></p> <p>f(2)=0</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/factor-and-remainder-theorem/esq_functions_remainder_th2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/factor-and-remainder-theorem/esq_functions_remainder_th2.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 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="474"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>It is given that <span class="math-tex">\(f(x)=x^3+ax^2+bx+8\)</span></p> <p>a. Given that <span class="math-tex">\(x^2-4\)</span> is a factor of <span class="math-tex">\(f(x)\)</span>, find the values of <strong><em>a </em></strong> and <strong><em>b</em></strong>.</p> <p>b. Factorize <span class="math-tex">\(f(x)\)</span> into a product of linear factors.</p> <p>c. Sketch the graph of <span class="math-tex">\(y=f(x)\)</span> labelling any stationary points and the x and y intercepts.</p> <p>d. <strong>Hence</strong> state the range of values of <strong><em>k</em></strong> for which <span class="math-tex">\(f(x)=k\)</span> has exactly one root.</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>c. Notice that you are allowed to use your calculator in this question. You should be able to find the stationary points from the calculator.</p> <p>d. Think of <span class="math-tex">\(f(x)=k\)</span> as a horizontal line. What values of <strong><em>k</em></strong> makes it cross the graph in only one place.<content></content></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/factor-and-remainder-theorem/esq_functions_remainder_th3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/factor-and-remainder-theorem/esq_functions_remainder_th3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="681"> <div class="panel-heading">MY PROGRESS</div> <div 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