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<h1 align="center">Factoring in Algebra</h1>
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<h2>Factors</h2>
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<p>Numbers have <a href="../numbers/factors-multiples.html">factors</a>:</p>
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<p align="center"><img src="../numbers/images/factor-2x3.svg" alt="factors 2x3=6" /></p>
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<p>And expressions (like <b>x<sup>2</sup>+4x+3</b>) also have factors:</p>
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<p align="center"><img src="images/polynomial-factors.svg" alt="factors" /></p>
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<h2>Factoring</h2>
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<p>Factoring (called "<b>Factorising</b>" in the UK) is the process of <b>finding the factors</b>:</p>
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<div class="def">
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<p> Factoring: Finding what to multiply together to get an expression.</p>
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</div>
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<p>It is like "splitting" an expression into a multiplication of simpler expressions.</p>
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<div class="example">
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<h3>Example: factor <span class="large">2y+6</span></h3>
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<p>Both 2y and 6 have a common factor of 2: </p>
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<ul>
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<li><span class="large" align="center">2y is 2 × y</span></li>
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<li><span class="large" align="center">6 is 2 × 3</span></li>
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</ul>
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<p>So we can factor the whole expression into: </p>
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<p align="center" class="large">2y+6 = 2(y+3)</p>
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<p>So <b>2y+6</b> has been "factored into" <b>2</b> and <b>y+3</b></p>
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</div>
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<p>Factoring is also the opposite of <a href="expanding.html">Expanding</a>:</p>
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<p align="center"><img src="images/expand-vs-factor.svg" alt="expand vs factor" /></p>
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<h2>Common Factor</h2>
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<p>In the previous example we saw that 2y and 6 had a common factor of <b>2</b></p>
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<p>But to do the job properly we need the <b>highest common factor</b>, including any variables</p>
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<div class="example">
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<h3>Example: factor <span class="large">3y<sup>2</sup>+12y</span></h3>
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<p>Firstly, 3 and 12 have a common factor of <span class="large">3</span>.</p>
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<p>So we could have:</p>
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<p align="center"><span class="large">3y<sup>2</sup>+12y = 3(y<sup>2</sup>+4y)</span></p>
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<p>But we can do better!</p>
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<p><span class="large">3y<sup>2</sup></span> and <span class="large">12y</span> also share the variable <span class="large">y</span>.</p>
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<p>Together that makes <span class="large">3y</span>:</p>
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<ul>
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<li><span class="large" align="center">3y<sup>2</sup> is 3y × y</span></li>
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<li><span class="large" align="center">12y is 3y × 4</span></li>
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</ul>
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<p> </p>
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<p>So we can factor the whole expression into:</p>
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<p align="center" class="large">3y<sup>2</sup>+12y = 3y(y+4)</p>
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<p> </p>
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<p>Check: <b>3y(y+4) = 3y × y + 3y × 4 = 3y<sup>2</sup>+12y</b></p>
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</div>
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<h2>More Complicated Factoring</h2>
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<h3>Factoring Can Be Hard !</h3>
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<p>The examples have been simple so far, but factoring <b>can</b> be very tricky.</p>
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<p> Because we have to figure <b>what got multiplied</b> to produce the expression we are given!</p>
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<p> </p>
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<p class="center"><img src="images/factor-cake.gif" alt="factoring cake" width="219" height="55" /><br>
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It is like trying to find which ingredients <br>
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went into a cake to make it so delicious.
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<br>
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<span class="larger">It can be hard to figure out!</span></p>
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<h3>Experience Helps</h3>
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<p>With more experience factoring becomes easier. </p>
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<div class="example">
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<h3>Example: Factor <b>4x<sup>2</sup> − 9</b></h3>
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<p>Hmmm... there don't seem to be any common factors.</p>
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<p>But knowing the <a href="special-binomial-products.html">Special Binomial Products</a> gives us a clue called the <b>"difference of squares"</b>: </p>
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<p align="center"><img src="images/diff-squares-example.gif" alt="difference of squares" width="130" height="99" /></p>
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<p align="center">Because <b>4x<sup>2</sup></b> is <b>(2x)<sup>2</sup></b>, and <b>9</b> is <b>(3)<sup>2</sup></b>, </p>
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<p>So we have:</p>
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<p align="center" class="larger">4x<sup>2</sup> − 9 = (2x)<sup>2</sup> − (3)<sup>2</sup></p>
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<p>And that can be produced by the difference of squares formula:</p>
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<p align="center" class="larger">(a+b)(a−b) = a<sup>2</sup> − b<sup>2</sup></p>
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<p class="center">Where <b>a</b> is 2x, and <b>b</b> is 3.</p>
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<p>So let us try doing that:</p>
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<p align="center" class="large">(2x+3)(2x−3) = (2x)<sup>2</sup> − (3)<sup>2</sup> = 4x<sup>2</sup> − 9</p>
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<p>Yes!</p><p> </p>
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<p>So the factors of <b>4x<sup>2</sup> − 9</b> are <b>(2x+3)</b> and <b>(2x−3)</b>:</p>
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<p align="center" class="large">Answer: 4x<sup>2</sup> − 9 = (2x+3)(2x−3)</p>
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</div>
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<p>How can you learn to do that? By getting lots of practice, and knowing "Identities"!</p>
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<p> </p>
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<h3>Remember these Identities</h3>
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<p>Here is a list of common "Identities" (including the <b>"difference of squares"</b> used above).</p>
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<p>It is worth remembering these, as they can make factoring easier.</p>
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<table border="0" align="center" cellpadding="6">
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<tr>
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<td colspan="3" align="center"><img src="images/expand-factor.svg" alt="factor expand" /></td>
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</tr>
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<tr>
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<td align="right"><span class="large">a<sup>2</sup> − b<sup>2</sup></span></td>
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<td><span class="large"> = </span></td>
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<td><span class="large">(a+b)(a−b)</span></td>
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</tr>
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<tr>
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<td align="right"><span class="large">a<sup>2</sup> + 2ab + b<sup>2</sup></span></td>
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<td><span class="large"> = </span></td>
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<td><span class="large">(a+b)</span><span class="large">(a+b)</span></td>
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</tr>
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<tr>
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<td align="right"><span class="large">a<sup>2</sup> − 2ab + b<sup>2</sup></span></td>
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<td><span class="large"> = </span></td>
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<td><span class="large">(a−b)</span><span class="large">(a−b)</span></td>
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</tr>
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<tr>
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<td align="right"><span class="large">a<sup>3</sup> + b<sup>3</sup></span></td>
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<td><span class="large"> = </span></td>
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<td><span class="large">(a+b)(a<sup>2</sup>−ab+b<sup>2</sup>)</span></td>
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</tr>
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<tr>
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<td align="right"><span class="large">a<sup>3</sup> − b<sup>3</sup></span></td>
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<td><span class="large"> = </span></td>
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<td><span class="large">(a−b)(a<sup>2</sup>+ab+b<sup>2</sup>)</span></td>
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</tr>
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<tr>
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<td align="right"><span class="large">a<sup>3</sup>+3a<sup>2</sup>b+3ab<sup>2</sup>+b<sup>3</sup></span></td>
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<td><span class="large"> = </span></td>
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<td><span class="large">(a+b)<sup>3</sup></span></td>
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</tr>
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<tr>
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<td align="right"><span class="large">a<sup>3</sup>−3a<sup>2</sup>b+3ab<sup>2</sup>−b<sup>3</sup></span></td>
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<td><span class="large"> = </span></td>
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<td><span class="large">(a−b)<sup>3</sup></span></td>
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</tr>
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</table>
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<p>There are many more like those, but those are the most useful ones.</p>
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<h2>Advice</h2>
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<p>The factored form is usually best.</p>
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<p>When trying to factor, follow these steps:</p>
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<ul>
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<div class="bigul">
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<li>"Factor out" any common terms</li>
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<li>See if it fits any of the identities, plus any more you may know</li>
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<li>Keep going till you can't factor any more</li>
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</div>
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</ul>
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<p>There are also Computer Algebra Systems (called "CAS") such as <i>Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce</i> and many more that are good at factoring.</p>
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<h2>More Examples</h2>
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<p>Experience does help, so here are more examples to help you on the way:</p>
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<div class="example">
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<h3>Example: w<sup>4</sup> − 16</h3>
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<p>An exponent of 4? Maybe we could try an exponent of 2:</p>
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<p align="center" class="larger">w<sup>4</sup> − 16 = (w<sup>2</sup>)<sup>2 </sup>− 4<sup>2</sup></p>
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<p>Yes, it is the difference of squares</p>
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<p align="center" class="larger">w<sup>4</sup> − 16 = (w<sup>2</sup><sup> </sup>+ 4)(w<sup>2</sup><sup> </sup>− 4)</p>
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<p>And "(w<sup>2</sup><sup> </sup>− 4)" is another difference of squares</p>
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<p align="center" class="larger">w<sup>4</sup> − 16 = (w<sup>2</sup><sup> </sup>+ 4)(w<sup> </sup>+ 2)(w<sup> </sup>− 2)</p>
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<p>That is as far as I can go (unless I use imaginary numbers)</p>
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</div>
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<div class="example">
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<h3>Example: 3u<sup>4</sup> − 24uv<sup>3</sup></h3>
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<p>Remove common factor "3u":</p>
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<p align="center" class="larger">3u<sup>4</sup> − 24uv<sup>3</sup> = 3u(u<sup>3</sup> − 8v<sup>3</sup>)</p>
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<p>Then a difference of cubes:</p>
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<p align="center" class="larger">3u<sup>4</sup> − 24uv<sup>3</sup> = 3u(u<sup>3</sup> − (2v)<sup>3</sup>)</p>
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<p align="center" class="larger"> = 3u(u−2v)(u<sup>2</sup>+2uv+4v<sup>2</sup>)</p>
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<p>That is as far as I can go.</p>
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</div>
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<div class="example">
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<h3>Example: z<sup>3</sup> − z<sup>2</sup> − 9z + 9</h3>
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<p>Try factoring the first two and second two separately:</p>
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<p align="center" class="larger">z<sup>2</sup>(z−1) − 9(z−1)</p>
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<p>Wow, <span class="larger">(z-1)</span> is on both, so let us use that:</p>
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<p align="center" class="larger">(z<sup>2</sup>−9)(z−1)</p>
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<p align="center" class="larger"> </p>
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<p>And <span class="larger">z<sup>2</sup>−9</span> is a difference of squares</p>
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<p align="center" class="larger">(z−3)(z+3)(z−1)</p>
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<p align="center" class="larger"></p>
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<p>That is as far as I can go.</p>
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</div>
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<p>Now get some more experience:</p>
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<div class="questions">
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<script type="text/javascript">getQ(338, 339, 2047, 2048, 2049, 178, 2050, 3177, 3178, 3179);</script>
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</div>
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<div class="related">
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<a href="index.html">Algebra Index</a> <a href="factoring-quadratics.html">Factoring Quadratics</a> </div>
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