Q: How do you factor the difference of two squares? How do you factor the perfect square trinomial? How do you factor the sum and difference of two cubes? Which of these three makes the most sense to

you? Explain why.

A: The other two special factoring formulas are two sides of the same coin: the sum and difference of cubes. [ These are the formulas:
a3 + b3 = (a + b)(a2 ? ab + b2)
a3 ? b3 = (a ? b)(a2 + ab + b2)
You'll learn in more advanced classes how they came up with these formulas. For now, just memorize them. First, notice that the terms in each factorization are the same; then notice that each formula

has only one "minus" sign. For the difference of cubes, the "minus" sign goes with the linear factor, a ? b; for the sum of cubes, the "minus" sign goes in the quadratic factor, a2 ? ab + b2. Some people use the mnemonic "SOAP" for the signs; the letters stand for "same" as the sign in the middle of the original expression, "opposite" sign, and "always positive".
a3 ? b3 = (a [same sign] b)(a2 [opposite sign] ab [always positive] b2)
Whatever method helps you best keep these formulas straight, do it, because you should not assume that you'll be given these formulas on the test. You really should know them. Note: The quadratic part of each cube formula does not factor, so don't attempt it.
When you have a pair of cubes, carefully apply the appropriate rule. By "carefully", I mean "using parentheses to keep track of everything, especially the negative signs". Here are some typical problems: Copyright ? Elizabeth Stapel 2000-2011 All Rights Reserved
Factor x3 ? 8
This is x3 ? 23, so I get:
x3 ? 8 = x3 ? 23
= (x ? 2)(x2 + 2x + 22)
= (x ? 2)(x2 + 2x + 4)
Factor 27x3 + 1
Remember that 1 can be regarded as having been raised to any power you like, so this is really (3x)3 + 13. Then I get:
27x3 + 1 = (3x)3 + 13
= (3x + 1)((3x)2 ? (3x)(1) + 12)
= (3x + 1)(9x2 ? 3x + 1)
Factor x3y6 ? 64
This is (xy2)3 ? 43, so I get:
x3y6 ? 64 = (xy2)3 ? 43
= (xy2 ? 4)((xy2)2 + (xy2)(4) + 42)
= (xy2 ? 4)(x2y4 + 4xy2 + 16) ]

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