Special Products and Factoring
SPECIAL PRODUCTS AND FACTORING STRATEGIES
When you learn to factor quadratics, there are three other formulas that they usually introduce at the same time. The first is the "difference of squares" formula.
Remember from your translation skills that "difference" means "subtraction". So a difference of squares is something that looks like x2 – 4. That's because 4 = 22, so you really have x2 – 22, a difference of squares. To factor this, do your parentheses, same as usual:
x2 – 4 = (x )(x )
You need factors of –4 that add up to zero, so use –2 and +2: x2 – 4 = (x – 2)(x + 2)
(Review Factoring Quadratics, if this example didn't make sense to you.)
Note that we had x2 – 22, and ended up with (x – 2)(x + 2). Differences of squares (something squared minus something else squared) always work this way:
For a2 – b2, do the parentheses:
( )( )
...put the first squared thing in front:
(a )(a )
...put the second squared thing in back:
(a b)(a b)
...and alternate the signs in the middles:
(a – b)(a + b)
Memorize this formula! It will come in handy later, especially when you get to rational expressions (polynomial fractions), and you'll probably be expected to know the formula for your next test.
Special Factoring: Factoring Sums and Differences of Cubes & Recognizing Perfect Squares
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
When you learn to factor quadratics, there are three other formulas that they usually introduce at the same time. The first is the "difference of squares" formula.
Remember from your translation skills that "difference" means "subtraction". So a difference of squares is something that looks like x2 – 4. That's because 4 = 22, so you really have x2 – 22, a difference of squares. To factor this, do your parentheses, same as usual:
x2 – 4 = (x )(x )
You need factors of –4 that add up to zero, so use –2 and +2: x2 – 4 = (x – 2)(x + 2)
(Review Factoring Quadratics, if this example didn't make sense to you.)
Note that we had x2 – 22, and ended up with (x – 2)(x + 2). Differences of squares (something squared minus something else squared) always work this way:
For a2 – b2, do the parentheses:
( )( )
...put the first squared thing in front:
(a )(a )
...put the second squared thing in back:
(a b)(a b)
...and alternate the signs in the middles:
(a – b)(a + b)
Memorize this formula! It will come in handy later, especially when you get to rational expressions (polynomial fractions), and you'll probably be expected to know the formula for your next test.
Special Factoring: Factoring Sums and Differences of Cubes & Recognizing Perfect Squares
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