2.7: Algebra Tips and Tricks Part III (Factoring)
Factoring
When factoring an expression like this:
\(x^2 - 8x + 15\)
The goal is to write this like \((x + a)(x + b)\) for some numbers \(a\) and \(b\), where \(a\) and \(b\) could be positive, negative, or zero. Since \((x + a)(x + b) = x^2 + (a + b)x + (ab)\) we see we need \(a + b = -8\) and \(ab = 15\). That way, when you foil it back out, you have \(x^2 - 8x + 15\). We see if \(a = -3\) and \(b = -5\), this works for both \(a + b = -8\) and \(ab = 15\). Thus,
\(x^2 - 8x + 15 = (x - 3)(x - 5)\)
Let’s do a couple more examples.
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Factor \(x^2 + 3x + 2\).
In this case we want \(a + b = 3\) and \(ab = 2\). \(a = 1\) and \(b = 2\) works, so \(x^2 + 3x + 2 = (x + 1)(x + 2)\).
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Factor \(x^2 + 5x - 84\).
This is a bit harder because the numbers are bigger, but we can still do it. We want \(a + b = 5\), and \(ab = -84\). We can see that \(84\) is \(12\) times \(7\). So if we have \(a = 12\) and \(b = -7\), then \(a + b = 5\) and \(ab = -84\). Hence \(x^2 + 5x - 84 = (x + 12)(x - 7)\).
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Factor \(x^2 - 64\).
In this case, we want \(a + b = 0\) and \(ab = -64\). But notice that this means \(a = -b\), and hence \(-a^2 = -64\), which means \(a^2 = 64\). That means \(a = 8\), so \(b = -8\) (or vice versa). Hence \(x^2 - 64 = (x + 8)(x - 8)\).