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