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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">Binomial Theorem HL</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/binomial-expansion/main_binomial_image-1.jpg" style="float: left; width: 100px; height: 100px;">The following page will help you with questions about the Binomial Theorem (or Binomial Expansion). The Binomial Theorem is used for expanding brackets in the form (a + b)<sup>n</sup> . Questions on this topic are usually short ones: you usually only have to find one term in an expansion rather than find the full expansion. In this case, it is important to be comfortable using combinations, but you should also be familiar with Pascal's triangle.</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>Expand <span class="math-tex">\((a+b)^n\)</span> using Pascal's triangle and <span class="math-tex">\(^nC_r\)</span> including for fractional and negative indices</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>Binomial Expansion - Spotting the Pattern</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="427"> <h3>Spotting the Pattern</h3> <p>At first look, the formula for the Binomial Theorem looks quite complicated</p> <p><span class="math-tex">\(\left(a+b\right)^{n}=a^{n}+\left(\begin{array}{c} n\\ 1 \end{array}\right) a^{n-1}b+...+\left(\begin{array}{c} n\\ r \end{array}\right)a^{n-r}b^{r}+...+b^{n} \)</span></p> <p>The formula is in the information booklet, so no need to try and lean it by heart! Applying it is not too difficult, but before we do that, it is important that we understand the pattern (or patterns) that lie behind it.</p> <p>The following video shows you the <strong>patterns</strong> behind the Binomial Theorem</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/214367256"></iframe></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>Using Pascal's Triangle</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="428"> <h3>Using Pascal's Triangle</h3> <p>The numbers in Pascal's triangle might well be familiar to you. Each number can be calculated by adding to the numbers immediately above it. As we saw in the previous video, the numbers in Pascal's triangle make up part of the pattern in the expansion of <span class="math-tex">\((a + b)^n\)</span></p> <p><em><img alt="" src="../../files/algebra/binomial-expansion/pascals_triangle.png" style="width: 305px; height: 281px;"></em></p> <p style="text-align: left;">We can use Pascal's triangle to help us expand brackets in the form <strong><span class="math-tex">\((a + b)^n\)</span></strong> . Watch the following video to see how to expand <span class="math-tex">\((1 +4x)^3\)</span>.</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/217406430"></iframe></div> </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>What are Combinations?</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="429"> <h3 style="text-align: left;">What are Combinations?</h3> <p>The numbers that appear in Pascal's triangle can also be found using combinations. HL students should know that <a href="../542/combinations-and-permutations.html" target="_blank" title="Combinations and Permutations">a combination</a> is a way of selecting items from a certain number of objects. We can write combinations in different ways <span class="math-tex">\(_{n} P_{r} \)</span> or <span class="math-tex">\(\binom {n} {r}\)</span></p> <p>The formula for combinations can be found in the information booklet <span class="math-tex">\(\binom {n} {r} = \frac{n!}{r!(n-r)!}\)</span></p> <p>The diagram below shows how each number in Pascal's triangle can be written as a combination.</p> <p><img alt="" src="../../files/algebra/binomial-expansion/pascals-triangle-image1.png" style="width: 640px; height: 281px;"></p> <p>Here's a video to explain in detail how we can find each number in Pascal's triangle with a combination.</p> <h3 style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="360" mozallowfullscreen="" src="https://player.vimeo.com/video/214911672" webkitallowfullscreen="" width="640"></iframe></h3> </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>Binomial Expansion using Combinations</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="430"> <h3 style="text-align: left;">Binomial Expansion using Combinations</h3> <p style="text-align: left;">Now we can use combinations to expand <span class="math-tex">\((a + b)^n\)</span>.</p> <p style="text-align: left;">The following video shows you how to expand <span class="math-tex">\({(1+2x)^4}\)</span> using combinations</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/214368097"></iframe></div> <p>The following video shows you how to expand <span class="math-tex">\({(2-3x)^3}\)</span> using combinations</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/214368575"></iframe></div> <p>The following video shows you how to expand <span class="math-tex">\((2x+{3 \over x})^5\)</span> using combinations</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/214369343"></iframe></div> </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>Binomial Expansion - Exam-style Questions</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="431"> <p>In the examination, it is unlikely that you are going to be asked to expand fully a set of brackets - it takes too long! Usually you are asked to find just one term. In the following video, we look at some typical exam-style questions:</p> <ol> <li> <p><em><strong>Consider the expansion of (1 + 2x)<sup>4</sup> .</strong></em><em><strong> Find the term containing x<sup>3</sup>.</strong></em></p> </li> <li> <p><em><strong>The third term in the expansion of (2x + p)<sup>6</sup> is 2160x<sup>4</sup>. Find the possible values of p.</strong></em></p> </li> <li> <p><em><strong>Find the term independent of x in the expansion of <span class="math-tex">\((x^3+ \frac{2}{x})^8\)</span></strong></em></p> </li> </ol> <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/215729333"></iframe></div> </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>Binomial Expansion - Exam-style Questions Difficult</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="432"> <p>In the following video, we look at a very challenging exam-style question:</p> <p><em><strong>Find the term x² in the expansion of <span class="math-tex">\((2-3x)^2(4- \frac{x}{2})^5\)</span></strong></em></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/217411643"></iframe></div> </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/algebra/binomial-expansion/revision-notes_binomial_expansionhl.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/binomial-expansion/revision-notes_binomial_expansionhl.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 short quiz to test your understanding of combinations</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#9079cebb"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="9079cebb"> <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%;"> Pascal's Triangle and Combinations <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="11-628-1732" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The 4th row of Pascal's triangle is</p><p style="margin-left: 40px;"><strong>1 4 <em>a</em> 4 1</strong></p><p>What is 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="6"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/algebra/binomial-expansion/pascals_triangle.png" style="width: 400px; height: 369px;"></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 5th row of Pascal's triangle is</p><p style="margin-left: 40px;"><strong>1 5 <em>a a</em> 5 1</strong></p><p>What is 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="10"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/algebra/binomial-expansion/pascals_triangle.png" style="width: 400px; height: 369px;"></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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Evaluate <span class="math-tex">\(\large^6C_2\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\large^6C_2\)</span> = <input type="text" style="height: auto;" data-c="15"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large^nC_r=\frac{n!}{(n-r)!\ r!}\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large^6C_2=\frac{6!}{4!\ 2!}=\frac{6\cdot 5}{2}=15\)</span></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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Evaluate <span class="math-tex">\(\large^8C_6\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\large^8C_6\)</span> = <input type="text" style="height: auto;" data-c="28"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large^nC_r=\frac{n!}{(n-r)!\ r!}\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large^8C_6=\frac{8!}{2!\ 6!}=\frac{8\cdot 7}{2}=28\)</span></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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Evaluate <span class="math-tex">\(\large^{10}C_3\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\large^{10}C_3\)</span> = <input type="text" style="height: auto;" data-c="120"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large^nC_r=\frac{n!}{(n-r)!\ r!}\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large^{10}C_3=\frac{10!}{7!\ 3!}=\frac{10\cdot 9 \cdot 8}{3\cdot 2}=120\)</span></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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Evaluate <span class="math-tex">\(\large^8C_8\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\large^8C_8\)</span> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large^nC_r=\frac{n!}{(n-r)!\ r!}\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large^8C_8=\frac{8!}{0!\ 8!}=1\)</span></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><img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Evaluate <span class="math-tex">\(\large^7C_6\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(\large^7C_6\)</span> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large^nC_r=\frac{n!}{(n-r)!\ r!}\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large^7C_6=\frac{7!}{1!\ 6!}=7\)</span></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><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Given that r>4, evaluate <strong><em>r</em></strong> , if <span class="math-tex">\(\large^6C_r=20\)</span></p></div><div class="q-answer"><p><strong><em>r</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large^nC_r=\frac{n!}{(n-r)!\ r!}\)</span></p><hr class="hidden-separator"><p>You can use a calculator to work this out using trial and error, or the table function</p><p><span class="math-tex">\(\large^6C_3=\frac{6!}{3!\ 3!}=\frac{6\cdot 5\cdot 4}{3\cdot 2}=20\)</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><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Evaluate <strong><em>r</em></strong> , if <span class="math-tex">\(\large^8C_r=56\)</span></p></div><div class="q-answer"><p><strong><em>r</em></strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large^nC_r=\frac{n!}{(n-r)!\ r!}\)</span></p><hr class="hidden-separator"><p>You can use a calculator to work this out using trial and error, or the table function</p><p><span class="math-tex">\(\large^8C_5=\frac{8!}{3!\ 5!}=\frac{8\cdot 7\cdot 6}{3\cdot 2}=56\)</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><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> Evaluate <strong><em>n</em></strong> , if <span class="math-tex">\(\large^nC_5=21\)</span></p></div><div class="q-answer"><p><strong><em>r</em></strong> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\large^nC_r=\frac{n!}{(n-r)!\ r!}\)</span></p><hr class="hidden-separator"><p>You can use a calculator to work this out using trial and error, or the table function</p><p><span class="math-tex">\(\large^7C_5=\frac{7!}{2!\ 5!}=\frac{7\cdot 6}{2}=21\)</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> </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> <hr class="hidden-separator"> <p>Here is a quiz that practises the Binomial expansion (a + b)<sup>n</sup> when n is a positive integer</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#d16c2a9c"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="d16c2a9c"> <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%;"> Binomial Expansion (new) <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="11-197-1732" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/algebra/binomial-expansion/10-rows-pascals-triangle.png" style="width: 512px; height: 219px;"></p><p>4 can be written as a combination <span class="math-tex">\(\left(\begin{array}{c} n\\ r \end{array}\right)\)</span>What are <strong>n </strong>and <strong>r?</strong></p></div><div class="q-answer">n = <input type="text" style="height: auto;" data-c="4 "> <span class="review"></span> r = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></div><div class="q-explanation"><p><span class="math-tex">\(\left(\begin{array}{c} 4\\ 1 \end{array}\right)\)</span>or <span class="math-tex">\(\left(\begin{array}{c} 4\\ 3 \end{array}\right)\)</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><img alt="" src="../../files/algebra/binomial-expansion/10-rows-pascals-triangle.png" style="width: 512px; height: 219px;"></p><p>21 can be written as a combination <span class="math-tex">\(\left(\begin{array}{c} n\\ r \end{array}\right)\)</span>What are <strong>n </strong>and <strong>r?</strong></p></div><div class="q-answer"><p>n= <input type="text" style="height: auto;" data-c="7"> <span class="review"></span> r = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\left(\begin{array}{c} 7\\ 2 \end{array}\right)\)</span>or <span class="math-tex">\(\left(\begin{array}{c} 7\\ 5 \end{array}\right)\)</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">\(\left(\begin{array}{c} 8\\ 2 \end{array}\right)\)</span>is equal to another combination <span class="math-tex">\(\left(\begin{array}{c} n\\ r \end{array}\right)\)</span>. What are <strong>n </strong>and <strong>r?</strong></p></div><div class="q-answer"><p>n= <input type="text" style="height: auto;" data-c="8"> <span class="review"></span> r= <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>Look at the symmetry of Pascal's triangle</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>Put these values in order of size from smallest to largest</p><p><span class="math-tex">\(\left(\begin{array}{c} 10\\ 8 \end{array}\right), \left(\begin{array}{c} 7\\ 5 \end{array}\right), \left(\begin{array}{c} 6\\ 2 \end{array}\right), \left(\begin{array}{c} 8\\ 2 \end{array}\right), \left(\begin{array}{c} 6\\ 3 \end{array}\right)\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\left(\begin{array}{c} 10\\ 8 \end{array}\right), \left(\begin{array}{c} 8\\ 2 \end{array}\right),\left(\begin{array}{c} 7\\ 5 \end{array}\right),\left(\begin{array}{c} 6\\ 2\end{array}\right), \left(\begin{array}{c} 6\\ 3 \end{array}\right)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\left(\begin{array}{c} 10\\ 8 \end{array}\right), \left(\begin{array}{c} 7\\ 5 \end{array}\right), \left(\begin{array}{c} 6\\ 2 \end{array}\right), \left(\begin{array}{c} 8\\ 2 \end{array}\right), \left(\begin{array}{c} 6\\ 3 \end{array}\right)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\left(\begin{array}{c} 6\\ 2 \end{array}\right), \left(\begin{array}{c} 8\\ 2 \end{array}\right), \left(\begin{array}{c} 6\\ 3 \end{array}\right), \left(\begin{array}{c} 7\\ 5 \end{array}\right), \left(\begin{array}{c} 10\\ 8 \end{array}\right)\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\left(\begin{array}{c} 6\\ 2\end{array}\right), \left(\begin{array}{c} 6\\ 3 \end{array}\right), \left(\begin{array}{c} 7\\ 5 \end{array}\right), \left(\begin{array}{c} 8\\ 2 \end{array}\right), \left(\begin{array}{c} 10\\ 8 \end{array}\right)\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\left(\begin{array}{c} 6\\ 2\end{array}\right)=15, \left(\begin{array}{c} 6\\ 3 \end{array}\right)=20, \left(\begin{array}{c} 7\\ 5 \end{array}\right)=21, \left(\begin{array}{c} 8\\ 2 \end{array}\right)=28, \left(\begin{array}{c} 10\\ 8 \end{array}\right)=45\)</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 <span class="math-tex">\({x^2}\)</span> term in the expansion <span class="math-tex">\({(1 - 2x)^6}\)</span> is <span class="math-tex">\({\textbf{a}x^2}\)</span> . Write down the value of <strong><em>a</em></strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="60"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\begin{pmatrix} 6 \\ 2 \end{pmatrix}(-2x)^21^4=15\times4x^2\times1\)</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 expansion of <span class="math-tex">\({(2 + 3x)^3}\)</span>is <span class="math-tex">\({\textbf{a}x^3+54x^2+\textbf{b}x+8}\)</span> . Find <em><strong>a</strong></em> and <em><strong>b</strong></em></p></div><div class="q-answer"><p>a =<input type="text" style="height: auto;" data-c="27"> <span class="review"></span></p>b = <input type="text" style="height: auto;" data-c="36"> <span class="review"></span></div><div class="q-explanation"><p><span class="math-tex">\(x^3\: term = \begin{pmatrix} 3 \\ 3 \end{pmatrix}(2)^0(3x)^3=1\times1\times(3x)^3 = 27x^3 \)</span></p><p><span class="math-tex">\(x \:term= \begin{pmatrix} 3 \\ 1 \end{pmatrix}(2)^2(3x)^1=3\times4\times(3x)^1 = 36x\)</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 expansion of <span class="math-tex">\(\)</span> <em><strong><span class="math-tex">\( (2x-{{ 1}\over x})^5\)</span></strong></em> is <span class="math-tex">\(32x^5 -80x^3+\textbf{a}x +{ \textbf{b}\over x}+ {10 \over x^3}-{1 \over x^5}\)</span>. Find <strong>a </strong>and <strong>b</strong>.</p></div><div class="q-answer"><p>a =<input type="text" style="height: auto;" data-c="80"> <span class="review"></span></p>b = <input type="text" style="height: auto;" data-c="-40"> <span class="review"></span></div><div class="q-explanation"><p><span class="math-tex">\(x\: term = \begin{pmatrix} 5 \\ 3 \end{pmatrix}(2x)^3({-1 \over x})^2=10\times8x^3\times{1 \over x^2} = 80x \)</span></p><p><span class="math-tex">\({1 \over x}\: term = \begin{pmatrix} 5 \\ 2 \end{pmatrix}(2x)^2({-1 \over x})^3=10\times4x^2\times{-1 \over x^3} = -40x \)</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>Find the <span class="math-tex">\(x^2\)</span> term in the expansion of <span class="math-tex">\(\)</span> <em><strong><span class="math-tex">\( (x^2+{{ 3}\over x})^4\)</span></strong></em>is <strong><span class="math-tex">\(\textbf{a}x^2\)</span></strong>. Find<strong> a</strong></p></div><div class="q-answer"> <input type="text" style="height: auto;" data-c="54"> <span class="review"></span></div><div class="q-explanation"><p>To find the <span class="math-tex">\(x^2\)</span> term you need to think about what powers you need to raise <span class="math-tex">\( x^2\)</span> and <span class="math-tex">\( {3\over x}\)</span></p><p>Try the different combinations</p><ul><li>4 and 0</li><li>3 and 1</li><li><strong>2 and 2</strong></li><li>1 and 3</li><li>0 and 4</li></ul><p><span class="math-tex">\(x^2\: term = \begin{pmatrix} 4 \\ 2 \end{pmatrix}(x^2)^2({3 \over x})^2=6\times x^4\times{9 \over x^2} = 54x^2 \)</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">\((1 + x)(2 + x^2)^3 = \textbf{a} + 8x + \textbf{b}x^2 + ...\)</span> Find <strong>a </strong>and <strong>b</strong></p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\((1 + x)(2 + x^2)^3 =(1 + x)(8 + 12x^2 + 6x^4+x^6)\)</span></p><p><img alt="" src="../../files/algebra/binomial-expansion/expansion_1.png" style="width: 340px; height: 43px;"></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 term independent of x in the expansion <span class="math-tex">\((x+ {2\over x})^6\)</span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="160"> <span class="review"></span></p></div><div class="q-explanation"><p>You can try to expand each term out, but you can save time by thinking about when the x term and <span class="math-tex">\({1 \over x}\)</span>will cancel out. <span class="math-tex">\(\left(\! \begin{array}{c} 6 \\ 3 \end{array} \!\right) x^3(\frac{2}{x})^3\)</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> </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> <hr class="hidden-separator"> <p>Here is another quiz that practises the Binomial expansion (a + b)<sup>n</sup> when n is a fraction or a negative value</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#34397efb"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="34397efb"> <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%;"> Binomial Expansion 2 <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="11-451-1732" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Complete the expansion</p><p><span class="math-tex">\((1+2x)^{-3}\)</span> = 1 + <strong><em>a</em></strong>x + 24x² + ...</p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span></p></div><div class="q-explanation"><p>Using the binomial expansion, we get</p><p>= 1 + (-3)(2x) + <span class="math-tex">\(\frac{(-3)(-2)}{2}(2x)^2\)</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>Using the binomial expansion, the expansion of <span class="math-tex">\((1-\frac{x}{2})^{-2}\)</span> is valid for |x| < <strong><em>a</em></strong></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="2"> <span class="review"></span></p></div><div class="q-explanation"><p>Valid for <span class="math-tex">\(|-\frac{x}{2}|<1\)</span></p><p>|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>In order to expand the following using the binomial expansion, we need to write it in the correct form</p><p><span class="math-tex">\((3-x)^{-2}=\frac{1}{a}(1-\frac{x}{b})^{-2}\)</span></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="9"> <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><span class="math-tex">\((3-x)^{-2}=3^{-2}(1-\frac{x}{3})^{-2}\)</span></p><p><span class="math-tex">\(=\frac{1}{9}(1-\frac{x}{3})^{-2}\)</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>Expand the following and work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p><p><strong><em><span class="math-tex">\(\frac{1}{1+x}=(1+x)^{-1}\)</span></em></strong></p><p>= 1 + <strong><em>a</em></strong>x + <strong><em>b</em></strong>x² + ...</p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>Using the binomial expansion, we get</p><p><span class="math-tex">\((1+x)^{-1}=1 + (-1)x + \frac{(-1)(-2)}{2}x^²+...\)</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>What is the value of <strong><em>a</em></strong> in the following expansion</p><p><span class="math-tex">\((1+a)^{\frac{1}{2}}\)</span> = 1 - 2x - 2x² + ...</p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-4x"> <span class="review"></span></p></div><div class="q-explanation"><p>Using the binomial expansion, we get</p><p><span class="math-tex">\((1+a)^{\frac{1}{2}}\)</span> <span class="math-tex">\(=1 + \frac{1}{2}a+\frac{(\frac{1}{2})(-\frac{1}{2})}{2}a^2+...\)</span></p><p><span class="math-tex">\(\frac{1}{2}a=-2x\)</span></p><p>a = -4x</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>What is the value of <strong><em>a</em></strong> in the following expansion</p><p><span class="math-tex">\((1+2x)^a=1-x+\frac{3x^2}{2}+...\)</span></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>Using the binomial expansion, we get</p><p>(1 + 2x)<sup>a</sup> = 1 + (a)2x + <span class="math-tex">\(\frac{a(a-1)}{2}(2x)^2+...\)</span></p><p>2ax = -x</p><p>a = - 0.5</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>What is the value of <strong><em>a</em></strong> in the following expansion</p><p>(1 + <strong><em>a</em></strong>x)<sup>-2</sup> = 1 - 6x + 27x² + ...</p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>Using the binomial expansion, we get</p><p>(1 + ax)<sup>-2</sup> = 1 + (-2)ax + <span class="math-tex">\(\frac{(-2)(-3)}{2}(ax)^2+...\)</span></p><p>-2ax = -6x</p><p>a = 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>What is the value of <strong><em>a</em></strong> in the following expansion</p><p><span class="math-tex">\(\sqrt{1-2x}=1-x+ax^2+...\)</span></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\sqrt{1-2x}=(1-2x)^\frac{1}{2}\)</span></p><p><span class="math-tex">\(=1+\frac{1}{2}(-2x)+\frac{\frac{1}{2}(-\frac{1}{2})}{2}(-2x)^2+...\)</span></p><p><span class="math-tex">\(\frac{\frac{1}{2}(-\frac{1}{2})}{2}(-2x)^2=ax^2\)</span></p><p><span class="math-tex">\(-\frac{1}{8}\cdot4x^2=ax^2\)</span></p><p>a = -0.5</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>Using the binomial expansion, the expansion of <span class="math-tex">\(\frac{1}{4-x}\)</span> is valid for |x| < <strong><em>a</em></strong></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="4"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{1}{4-x}=(4-x)^{-1}\)</span></p><p><span class="math-tex">\(=4^{-1}(1-\frac{x}{4})^{-1}\)</span></p><p>Valid for <span class="math-tex">\(|-\frac{x}{4}|<1\)</span></p><p>Valid for |x| < 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>What is the value of <strong><em>a</em></strong> in the following expansion</p><p><span class="math-tex">\((2+ax)^{-1}=\frac{1}{2}+\frac{3}{2}x+\frac{9}{2}x^2+...\)</span></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\((2+ax)^{-1}=2^{-1}(1+\frac{ax}{2})^{-1}\)</span></p><p><span class="math-tex">\(=\frac{1}{2}(1+(-1)\frac{ax}{2}+\frac{(-1)(-2)}{2}(\frac{ax}{2})^2+...)\)</span></p><p><span class="math-tex">\(-\frac{ax}{4}=\frac{3}{2}x\)</span></p><p>a = -6</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="437"> <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/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL 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 values of the third row of Pascal's triangle are given below</p> <p><img alt="" src="../../files/algebra/binomial-expansion/esq1.jpg" style="width: 400px; height: 66px;"></p> <p>a) Write down the values in the fourth row of Pascal's triangle</p> <p>b) Hence or otherwise, find the term in x² in the expansion of <span class="math-tex">\((3x+2)^4\)</span></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">The fourth row of Pascal's triangle give the coefficient to multiply by (3x)<sup>2</sup> (2)<sup>2</sup><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/binomial-expansion/esq_binomial-expansion1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/binomial-expansion/esq_binomial-expansion1.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="436"> <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/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL 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>a) Expand (x - 3)<sup>4</sup> and simplify your result</p> <p>b) Hence find the x<sup>3</sup> term in (x - 2)(x - 3)<sup>4</sup> .</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">No calculator here...you may find it easier to write out the fourth row of Pascal's Triangle<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/binomial-expansion/esq_binomial-expansion2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/binomial-expansion/esq_binomial-expansion2.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-has-border panel-expandable panel-default"> <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="435"> <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/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>Find the term independent of x in the expansion <span class="math-tex">\((2x- \frac{3}{x^2})^6\)</span></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>The term independent of <strong><em>x </em></strong>is the term without <strong><em>x </em></strong>in it.</p> <p><content>Consider different possible powers of the two parts <span class="math-tex">\((2x)\)</span>and <span class="math-tex">\((-\frac{3}{x^2})\)</span> </content>.</p> <p>Remember these powers should add up to 6.</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/binomial-expansion/esq_binomial-expansion3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/binomial-expansion/esq_binomial-expansion3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="845"> <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>a) Use the binomial theorem to expand <span class="math-tex">\(\sqrt{4-9x}\)</span> in ascending powers of x up to and including x<sup>3</sup></p> <p>b) State the value of x for which the expansion is valid.</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">Write as <span class="math-tex">\(4^{\frac{1}{2}}(1-\frac{9x}{4})^{\frac{1}{2}}\)</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/algebra/binomial-expansion/esq_binomial-expansion6.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/binomial-expansion/esq_binomial-expansion6.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <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="434"> <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/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>The x term in the expansion <span class="math-tex">\((4+2x)^3(2+ax)^4\)</span> is -4608x</p> <p>Find <strong><em>a</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>There is no need to expand<span class="math-tex">\((4+2x)^3(2+ax)^4\)</span> completely!</p> <p><content>Consider how you will get the <strong><em>x</em></strong> term</content></p> <p><img alt="" src="../../files/algebra/binomial-expansion/hintesq4.jpg" style="width: 300px; height: 69px;"></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/binomial-expansion/esq_binomial-expansion4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/binomial-expansion/esq_binomial-expansion4.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-has-border panel-expandable panel-default"> <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="846"> <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>In the expansion <span class="math-tex">\((1+bx)^n\)</span>, the coefficient of the x term is -6 and the coefficient of the x² term is 27.</p> <p>Work out <strong><em>b</em></strong> and <strong><em>n</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>Use the Binomial expansion to work out the x and x² terms.</p> <p>You will need to solve simultaneous equations to find <strong><em>b</em></strong> and <strong><em>n</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/binomial-expansion/esq_binomial-expansion7.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/binomial-expansion/esq_binomial-expansion7.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 7</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="847"> <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>a) Write <span class="math-tex">\(f(x)=\frac{2+x}{1+x-2x^2}\)</span> as the sum of partial fractions</p> <p>b) Find the binomial expansion of <strong><em>f(x)</em></strong> in ascending powers of <strong><em>x</em></strong> up to and including <strong><em>x²</em></strong></p> <p>c) State the values of <strong><em>x</em></strong> for which the series is valid</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>a) <span class="math-tex">\(f(x)=\frac{A}{1-x}+\frac{B}{1+2x}\)</span></p> <p><content>b) Write an expansion for the 2 separate fractions, then add the results together</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/algebra/binomial-expansion/esq_binomial-expansion8.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/binomial-expansion/esq_binomial-expansion8.pdf" width="640"></iframe></p> </section> <h4> </h4> </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 8</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="433"> <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/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><span></span><span class="math-tex">\(\large(\textbf{a}+2x)^3(4-x)^4 = 6912 + \textbf{b}x +...\)</span><span></span></p> <p>Find <strong>a</strong> and <strong>b</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>There is no need to expand completely <span class="math-tex">\((\textbf{a}+2x)^3(4-x)^4\)</span></p> <p><content></content><img alt="" src="../../files/algebra/binomial-expansion/hintesq5.jpg" style="width: 400px; height: 69px;"></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/binomial-expansion/esq_binomial-expansion5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/binomial-expansion/esq_binomial-expansion5.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="1732"> <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>Binomial Theorem HL</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 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