Find a common monomial factor. = 2y 3 (y + 3)(y 3) Difference of two squares
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1 EXAMPLE 1 Find a common monomial factor Factor the polynomial completely. a. x 3 + 2x 2 15x = x(x 2 + 2x 15) Factor common monomial. = x(x + 5)(x 3) Factor trinomial. b. 2y 5 18y 3 = 2y 3 (y 2 9) Factor common monomial. = 2y 3 (y + 3)(y 3) Difference of two squares c. 4z 4 16z z 2 = 4z 2 (z 2 4z + 4) Factor common monomial. = 4z 2 (z 2) 2 Perfect square trinomial
2 EXAMPLE 2 Factor the sum or difference of two cubes Factor the polynomial completely. a. x = x Sum of two cubes = (x + 4)(x 2 4x + 16) b. 16z 5 250z 2 = 2z 2 (8z 3 125) Factor common monomial. = 2z 2 (2z) Difference of two cubes = 2z 2 (2z 5)(4z z + 25)
3 GUIDED PRACTICE for Examples 1 and 2 Factor the polynomial completely. 1. x 3 7x x SOLUTION x 3 7x x = x 3 7x x = x( x 2 7x + 10) = x( x 5 )( x 2 ) Factor common monomial. Factor trinomial.
4 GUIDED PRACTICE for Examples 1 and y 5 75y 3 SOLUTION 3y 5 75y 3 = 3y 3 (y 2 25) = 3y 3 (y 5)( y + 5 ) Factor common monomial. Difference of two squares
5 GUIDED PRACTICE for Examples 1 and b b 2 SOLUTION 16b b 2 = 2b 2 (8b ) = 2b 2 (2b + 7)(4b 2 14b + 49 ) Factor common monomial. Difference of two cubes
6 GUIDED PRACTICE for Examples 1 and 2 4. w 3 27 SOLUTION w 3 27 = w 3 (3) 3 = (w 3)(w 2 + 3w + 9) Difference of two cubes
7 EXAMPLE 3 Factor by grouping Factor the polynomial x 3 3x 2 16x + 48 completely. x 3 3x 2 16x + 48 = x 2 (x 3) 16(x 3) Factor by grouping. = (x 2 16)(x 3) Distributive property = (x + 4)(x 4)(x 3) Difference of two squares
8 EXAMPLE 4 Factor polynomials in quadratic form Factor completely: (a) 16x 4 81 and (b) 2p p p 2. a. 16x 4 81 = (4x 2 ) Write as difference of two squares. = (4x 2 + 9)(4x 2 9) Difference of two squares = (4x 2 + 9)(2x + 3)(2x 3) Difference of two squares b. 2p p p 2 = 2p 2 (p 6 + 5p 3 + 6) = 2p 2 (p 3 + 3)(p 3 + 2) Factor common monomial. Factor trinomial in quadratic form.
9 GUIDED PRACTICE for Examples 3 and 4 Factor the polynomial completely. 5. x 3 + 7x 2 9x 63 SOLUTION x 3 + 7x 2 9x 63 = x 2 (x + 7) 9(x + 3) Factor by grouping. = (x 2 9)(x + 7) Distributive property = (x + 3)(x 3) (x + 7) Difference of two squares
10 GUIDED PRACTICE for Examples 3 and g SOLUTION a. 16g = (4g 2 ) Write as difference of two squares. = (4g )(4g 2 25) Difference of two squares = (4g )(2g + 5)(2g 5) Difference of two squares
11 GUIDED PRACTICE for Examples 3 and t 6 20t t 2 SOLUTION 4t 6 20t t 2 = 4t 2 (t 4 5t 2 6t) = 4t 2 (t 2 3)(t 2 2 ) Factor common monomial. Factor trinomial in quadratic form.
12 EXAMPLE 5 Standardized Test Practice SOLUTION 3x x = 18x 3 3x 5 18x x = 0 3x(x 4 6x 2 + 5) = 0 Write original equation. Write in standard form. Factor common monomial.
13 EXAMPLE 5 Standardized Test Practice 3x(x 2 1)(x 2 5) = 0 3x(x + 1)(x 1)(x 2 5) = 0 x = 0, x = 1, x = 1, x = 5,or x = 5 Factor trinomial. Difference of two squares Zero product property ANSWER The correct answer is D.
14 GUIDED PRACTICE for Example 5 Find the real-number solutions of the equation. 8. 4x 5 40x x = 0 4x 5 40x x = 0 4x(x 4 10x 2 + 9) = 0 4x(x 2 1)(x 2 9) = 0 4x(x + 1)(x 1)( x 3)(x + 3) = 0 Write original equation. Factor common monomial. Factor trinomial. Difference of two squares x = 0, x = 1, x = 1, x = 3,or x = 3 Zero product property ANSWER 3, 3, 1, 1, 0
15 GUIDED PRACTICE for Example x x = 14x 3 2x x = 14x 3 2x 5 14x x = 0 2x(x 4 7x ) = 0 2x(x 2 4)(x 2 3) = 0 2x(x + 2)(x 2)( x 2 3) = 0 Write original equation. Write in standard form. Factor common monomial. Factor trinomial. Difference of two squares ANSWER 3, 3, 2, 0, 2
16 GUIDED PRACTICE for Example x x 2 = 6x 4 27x x 2 = 6x 4 Write original equation. 6x 4 27x 3 15x 2 = 0 3x 2 (2x 2 9x 5) = 0 Write in standard form. Factor common monomial. ANSWER 0,
17 EXAMPLE 6 Solve a polynomial equation City Park You are designing a marble basin that will hold a fountain for a city park. The basin s sides and bottom should be 1 foot thick. Its outer length should be twice its outer width and outer height. What should the outer dimensions of the basin be if it is to hold 36 cubic feet of water?
18 EXAMPLE 6 SOLUTION Solve a polynomial equation 36 = (2x 2)(x 2)(x 1) 0 = 2x 3 8x x 40 0 = 2x 2 (x 4) + 10(x 4) Write equation. Write in standard form. Factor by grouping. 0 = (2x )(x 4) Distributive property ANSWER The only real solution is x = 4. The basin is 8 ft long, 4 ft wide, and 4 ft high.
19 GUIDED PRACTICE for Example What if? In Example 6, what should the basin s dimensions be if it is to hold 128 cubic feet of water and have outer length 6x, width 3x, and height x? ANSWER length: about15.66 ft; width: about ft; height: about 2.61 ft
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