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      <text>3.6.2.4_Example_Express a fraction in terms of partial fractions</text>
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<h4 class="HELM_example">Example</h4>
<p>Express \(\dfrac{7x+10}{ 2x^2+5x+3}\) in terms of partial fractions.</p>
<h4 class="HELM_solution">Solution</h4>
<p>Note that this fraction is proper. The denominator is factorised to give <span class="math inline">\((2x+3)(x+1)\)</span>. Each of the linear factors produces a partial fraction. The factor <span class="math inline">\(2x+3\)</span> produces a partial fraction of the form <span class="math inline">\(\dfrac{A}{2x+3}\)</span> and the factor <span class="math inline">\(x+1\)</span> produces a partial fraction <span class="math inline">\(\dfrac{B}{x+1}\)</span>, where <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> are constants which we need to find. We write </p>
<p>\[\dfrac{7x+10}{(2x+3)(x+1)} = \dfrac{A}{2x+3} + \dfrac{B}{x+1}\]</P
<p>By multiplying both sides by <span class="math inline">\((2x+3)(x+1)\)</span> we obtain</p> 
<p>\[7x+10 = A(x+1) + B(2x+3)…(*)\] </p>
<p>We may now let <span class="math inline">\(x\)</span> take <b>any value we choose</b>. By an appropriate choice we can simplify the right-hand side. Let <span class="math inline">\(x=-1\)</span> because this choice eliminates <span class="math inline">\(A\)</span>. We find <span class="math display">\[\begin{aligned}
7(-1) + 10 &amp;= A(0) + B(-2+3)\\
3&amp;=B
\end{aligned}\]</span> so that the constant <span class="math inline">\(B\)</span> must equal 3. The constant <span class="math inline">\(A\)</span> can be found either by substituting some other value for <span class="math inline">\(x\)</span> or alternatively by ‘equating coefficients’. Observe that, by rearranging the right-hand side, Equation <b>(*)</b> can be written as </p>
<p>\[7x+10 = (A+2B)x + (A+3B)\]</p> 
<p>Comparing the coefficients of <span class="math inline">\(x\)</span> on both sides we see that <span class="math inline">\(7=A+2B\)</span>. We already know <span class="math inline">\(B=3\)</span> and so <span class="math display">\[\begin{aligned}
7&amp;=A + 2(3)\\
&amp;= A+ 6\end{aligned}\]</span> from which <span class="math inline">\(A=1\)</span>. We can therefore write </p>
<p>\[\dfrac{7x+10}{2x^2+5x+3} = \dfrac{1}{2x+3} + \dfrac{3}{x+1}\]</p> 
<p>We have succeeded in expressing the given fraction as the sum of its partial fractions. The result can always be checked by adding the fractions on the right.</p>]]></text>
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