<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="description">
    <name>
      <text>1.3.3.1</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h3> 3. Removing brackets from expressions of the form \( (a+b)(c+d) \)</h3>

Sometimes it is necessary to consider two bracketed terms multiplied together. In the expression
\( (a + b)(c + d) \), by regarding the first bracket as a single term we can use the result given previously to write it as \( (a + b)c + (a + b)d \). Removing the brackets from each of these terms produces
\( ac + bc + ad + bd \). More concisely:
<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>\[ (a + b)(c + d) = (a + b)c + (a + b)d = ac + bc + ad + bd. \].</p>
</div>

We see that each term in the first bracketed expression multiplies each term in the second bracketed
expression.<p><br></p><p>&nbsp;

</p><hr>
<h4 class="HELM_example">Example</h4>
<p> Remove the brackets from \( (3 + x)(2 + y) \).</p>
<h4 class="HELM_solution">Solution</h4>
<p>We find \[ (3 + x)(2 + y) = (3 + x)(2) + (3 + x)y \]
\[ = (3)(2) + (x)(2) + (3)(y) + (x)(y) = 6 + 2x + 3y + xy. \]</p>

<h4 class="HELM_example">Example</h4>
<p> Remove the brackets from \( (3x + 4)(x + 2) \) and simplify your result.</p>
<h4 class="HELM_solution">Solution</h4>
<p>
\[ (3x + 4)(x + 2) = (3x + 4)(x) + (3x + 4)(2) \]
\[ = 3x2 + 4x + 6x + 8 = 3x2 + 10x + 8. \]</p>

<h4 class="HELM_example">Example</h4>
<p> Remove the brackets from \( (a + b)^2 \) and simplify your result.</p>
<h4 class="HELM_solution">Solution</h4>
<p>When a quantity is squared it is multiplied by itself. Thus \[ (a + b)^2 = (a + b)(a + b) = (a + b)a + (a + b)b \] \[ = a^2 + ba + ab + b^2 = a^2 + 2ab + b^2. \]</p><br>

<div class="HELM_keypoint">
<h4>Key Point</h4><p>
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
\[ (a - b)^2 = a^2 - 2ab + b^2 \]</p>
</div>

<hr>
<h4 class="HELM_example">Example</h4>
<p> Explain the distinction between \( (x + 3)(x + 2)\) and \( x + 3(x + 2). \)</p>
<h4 class="HELM_solution">Solution</h4>
<p style="">In the first expression removing the brackets we find
\[ (x + 3)(x + 2) = x^2 + 3x + 2x + 6 \]
\[ = x^2 + 5x + 6. \]

In the second expression we have
\[ x + 3(x + 2) = x + 3x + 6 = 4x + 6. \]
Note that in the second expression the term \( (x + 2) \) is only multiplied by \(3\) and not by \(x\).</p>

<hr>
<h4 class="HELM_example">Example</h4>
<p> Remove the brackets from \( (s^2 + 2s + 4)(s + 3) \).</p>
<h4 class="HELM_solution">Solution</h4>
<p>Each term in the first bracket must multiply each term in the second. Working through all combinations systematically we have
\[ (s^2 + 2s + 4)(s + 3) = (s^2 + 2s + 4)(s) + (s^2 + 2s + 4)(3) \]
\[ = s^3 + 2s^2 + 4s + 3s^2 + 6s + 12 \]
\[ = s^3 + 5s^2 + 10s + 12.\]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text/>
    </generalfeedback>
    <defaultgrade>0</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
  </question>
</quiz>
