Polynomials and Polynomial Functions *Chapter 5 Polynomials and Polynomial Functions
Chapter Sections 5.1 – Addition and Subtraction of Polynomials 5.2 – Multiplication of Polynomials 5.3 – Division of Polynomials and Synthetic Division 5.4 – Factoring a Monomial from a Polynomial and Factoring by Grouping 5.5 – Factoring Trinomials 5.6 – Special Factoring Formulas 5.7-A General Review of Factoring 5.8- Polynomial Equations Chapter 1 Outline
5.6 Special Factoring Formulas a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) 5.6 Special Factoring Formulas a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 (a + b)(a – b) = a2 – b2 1. Factor perfect square trinomials. 2. Factor a difference of squares. 3. Factor a difference of cubes. 4. Factor a sum of cubes.
Difference of Two Squares (a + b)(a – b) = a2 – b2 Example 1: a.) Factor x2 – 16. x2 – 16 = x2 – 42 = (x + 4)(x – 4) b.) Factor 25x2 – 36y2. 25x2 – 36y2 = (5x)2 – (6y)2 = (5x + 6y)(5x – 6y)
Example 2: (a + b)(a – b) = a2 – b2 a)Factor. 9x2 – 16y2 Solution b)16n4 – 25 Solution 9x2 – 16y2 = (3x)2 – (4y)2 . a2 – b2 = (a + b)(a – b) (3x)2 – (4y)2 = (3x + 4y)(3x – 4y) = (4n2 + 5)(4n2 – 5)
Factor Perfect Square Trinomials a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 Example 3: Factor a.) x2 – 8x + 16. – 2(x)(4) = – 8x x2 – 8x + 16 = (x – 4)2 b) 16x2 – 56x + 49 =(4x – 7)2
Example 4: Factor. a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 a) 9a2 + 6a + 1 b) 9x2 – 6xy + y2 = (3x – y)2 c) Answer 9a2 + 6a + 1 = (3a + 1)2 The square root of 9a2 is 3a. The square root of 1 is 1. Twice the product of 3a and 1 is (2)(3a)(1) = 6a, which is the middle term. a = 3x and b = y. a = 3a and b = 7.
Sum of Two Cubes Example 5: Factor the sum of cubes x3 + 64. a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) a3 + b3 = (a + b) (a2 – ab + b2) Example 5: Factor the sum of cubes x3 + 64.
Example 6: Factor. a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) Example 6: Factor. a) 125x3 + 64 c) 6x +162xy3 a3 + b3 = (a + b) (a2 a b + b2) 125x3 + 64 = (5x)3 + (4)3 = (5x + 4)((5x)2 (5x)(4) + (4)2) = (5x + 4)(25x2 20x + 16) 6x +162xy3 = 6x(1 + 27y3) = 6x[13 + (3y)3] = 6x(1 + 3y)((1)2 – (1)(3y) + (3y)2) = 6x(1 + 3y)(1 – 3y + 9y2)
Difference of Two Cubes a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) Example 7: Factor. 125x3 – 27 Solution a3 – b3 = (a – b) (a2 + a b + b2) 125x3 – 27 = (5x)3 – (3)3 = (5x – 3)((5x)2 + (5x)(3) + (3)2) = (5x – 3)(25x2 + 15x + 9) Note: The trinomial may seem like a perfect square. However, to be a perfect square, the middle term should be 2ab. In this trinomial, we only have ab, so it cannot be factored.
Difference of Two Cubes a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) Difference of Two Cubes Example 8: a.) Factor 27x3 – 8y6.
Example 9: a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2) Factor. Solution
Helpful Hint for Factoring When factoring the sum or difference of two cubes, the sign between the terms in the binomial factor will be the same as the sign between the terms. The sign of the ab term will be the opposite of the sign between the terms of the binomial factor. The last term in the trinomial will always be positive. a3 + b3 = (a + b) (a2 – ab + b2) same sign opposite sign always positive a3 – b3 = (a – b) (a2 + ab + b2) same sign opposite sign always positive