Section 1.5: Factoring Special Products
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1 Objective: Identify and factor special products including a difference of two perfect squares, perfect square trinomials, and sum and difference of two perfect cubes. When factoring there are a few special products that, if we can recognize them, help us factor polynomials. DIFFERENCE OF TWO PERFECT SQUARES When multiplying special products, we found that a sum of a binomial and a difference of a binomial could multiply to a difference of two perfect squares. Here, we will use this special product to help us factor. Difference of Two Perfect Squares: a b a b a b ( ( Eample 1. Factor completely. 16 Epress each term as the square of a monomial ( ( 4 Apply the difference of two perfect squares formula: Here, a and b 4 ( 4( 4 Eample. Factor completely. 6 y Epress each term as the square of a monomial (6 ( y Apply the difference of two perfect squares formula: Here, a 6 and b y (6 y(6 y Eample. Factor completely. 9a 5b Epress each term as the square of a monomial ( a (5b Apply the difference of two perfect squares formula: Here, a a and b 5b (a 5 b(a 5 b Page 9
2 PERFECT SQUARE TRINOMIAL Another special case involves the perfect square trinomial. We had a shortcut for squaring a binomial, which can be reversed to help us factor a perfect square trinomial. Perfect Square Trinomial: a ab b ( a b a ab b ( a b If we do not recognize a perfect square trinomial at first glance, we use the ac method. If we get two of the same numbers, we know we have a perfect square trinomial. Then we can factor using the square roots of the first and last terms, and the sign from the middle term. Eample 4. Factor completely. 6 9 Multiply to 9, sum to 6 Numbers are and, the same; a perfect square trinomial Use square roots from first and last terms and sign from middle term ( Eample 5. Factor completely. 40y 5y Multiply to 100, sum to 0 Numbers are 10 and 10, the same; perfect square trinomial Use square roots from first and last terms and sign from middle term ( 5 y SUM OR DIFFERENCE OF TWO PERFECT CUBES Another special case involves the sum or difference of two perfect cubes. The sum and the difference of two perfect cubes have very similar factoring formulas: Sum of Two Perfect Cubes: a b ( a b( a ab b Difference of Two Perfect Cubes: a b ( a b( a ab b Page 0
3 Start by epressing each term as the cube of a monomial. Use these results to determine the factored form of the epression. Comparing the formulas, you may notice that the only difference is the signs between the terms. One way to keep these two formulas straight is to think of SOAP. S stands for Same sign as the original polynomial. If we have a sum of two perfect cubes, we add first; if we have a difference of two perfect cubes we subtract first. O stands for Opposite sign. If we have a sum, then subtraction is the second sign; a difference has addition for the second sign. AP stands for Always Positive. The last term for both formulas has an addition sign. The following eamples demonstrate factoring the sum or difference of two perfect cubes. Eample 6. Factor completely. m 7 Epress each term as the cube of a monomial ( m ( Apply the difference of two perfect cubes formula ( m ( m m 9 ; Use SOAP to fill in signs ( m ( m m 9 Eample 7. Factor completely. 15 p 8r Epress each term as the cube of a monomial (5 p (r Apply the sum of two perfect cubes formula (5p r (5p 10r 4 r ; Use SOAP to fill in signs (5p r (5p 10 pr 4r The previous eample illustrates an important point. When we fill in the trinomial s first and last terms, we square the monomials 5p and r. So, our squared terms in the second set of parentheses are 5p 5p 5p and r r 4r. Notice that when done correctly, both the number and the variable are squared. Sometimes students forget to square both the number and the variable. Often after factoring a sum or difference of cubes, students want to factor the second factor, the trinomial, further. As a general rule, this factor will always be prime (unless there is a GCF that should have been factored before applying the appropriate perfect cubes rule. SUMMARY OF FACTORING SPECIAL PRODUCTS The following table summarizes all of the methods that we can use to factor special products: Page 1
4 FACTORING SPECIAL PRODUCTS Difference of Squares: Sum of Squares: Perfect Square Trinomial: a b a b a b prime ( ( a ab b ( a b a ab b ( a b Sum of Cubes: Difference of Cubes: a b ( a b( a ab b a b ( a b( a ab b FACTORING USING MORE THAN ONE STRATEGY As always, when factoring special products it is important to check for a GCF first. Only after checking for a GCF should we be using the special products. This process is shown in the following eamples. Eample 8. Factor completely. 7 GCF is ; factor from each term (6 1 Difference of two perfect squares: (6 1 (6 1 6 ( 6 and 1 (1 Eample 9. Factor completely. 48 y 4y y GCF is y ; factor from each term y( Multiply to 16, sum to 8 Numbers are 4 and 4, the same; perfect square trinomial Use square roots from first and last terms and sign from middle term y(4 1 Eample 10. Factor completely a b 54ab GCF is ab ; factor from each term ab (64a 7 b Sum of two perfect cubes: 64a ( 4a and 7b ( b ab (4a b (16a 1ab 9 b Page
5 Practice Eercises Factor completely v p k 4 5 4k 4 5p 10 p 1 1 5a 0ab 9b 8y 16y 6 4v 1 6 4a 0ab 5b y y a n y 98a 50b 4m 64n a k 19 n 0 a 1 4k n y 18y 00y 5y u a y m 18 0 n y Page
6 ANSWERS to Practice Eercises 1 ( 7( 7 ( ( ( v 5( v 5 4 (1 (1 5 ( p ( p 6 (v 1( v 1 7 (8 y(8 y 8 (a 1( a 1 9 prime 10 ( ( 11 5( n ( n 1 4( ( 1 5(5 9y 14 (7a 5 b(7a 5 b ( m 6 ( a 1 ( k ( ( n 4 ( 1 n ( k (5p 1 ( 1 (5a b ( 4 y (a 5 b 7 prime ( y 5( y ( ( 4 ( 1 4( 4 6 ( 1 4( 4 6 ( 4 ( (6 u(6 6 u u (5 6(5 0 6 (5a 4(5 a 0a 16 (4 ( (4 y(16 1y 9y 4(m n(4 m 6mn 9n (5y( 9 15y 5y Page 4
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