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

5.7 A General Review of Factoring a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2

Check: Multiplying the factors. Factoring a Polynomial 1. GCF first factor out any monomial GCF 2. Consider Number of Terms 3. 4 terms: Factor by grouping 4. 3 terms: a) perfect square: a2 + 2ab + b2 = (a + b)2 Or a2  2ab + b2 = (a  b)2. b) Not a perfect square: c) x2 + bx + c, find two factors of c whose sum = b & (x + 1st # )(x + 2nd # ) d) ax2 + bx + c, where a  1, Use trial and error. Or, find 2 factors of ac whose sum = b; write these factors as coefficients of two like terms that, when combined, equal bx; and then factor by grouping. 5. 2 terms: a) A difference of squares: a2 – b2 = (a + b)(a – b) b) A sum of cubes: a3 + b3 = (a + b)(a2 – ab + b2). c) A difference of cubes: a3 – b3 = (a – b)(a2 + ab + b2). Check: Multiplying the factors.

Examples 1: Factor. a.) 2x4 – 50x2 b.) 24x2 – 6xy + 40xy – 10y2 a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 Factor. a.) 2x4 – 50x2 b.) 24x2 – 6xy + 40xy – 10y2 c.) 12x2 – 8x – 15 = (6x + 5)(2x – 3)

Examples 2: Factor. a) 5x3 – 10x2 – 120x b) 12y5 + 84y3 c) 8a4 – 72n2 a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 Examples 2: Factor. a) 5x3 – 10x2 – 120x b) 12y5 + 84y3 c) 8a4 – 72n2 Solution = 5x(x2 – 2x – 24) = 5x(x + 4)(x – 6) = 8(a4 – 9n2) = 12y3(y2 + 7) = 8(a2 – 3n)(a2 + 3n)

Examples 3: Factor a)150x3y – 120x2y2 + 24xy3 Solution a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 a)150x3y – 120x2y2 + 24xy3 Solution 6xy(25x2 – 20xy + 4y2 ) = 6xy(5x – 2y)2 b)x5 – 2x3 – 27x2 + 54 = x3(x2 – 2) – 27(x2 – 2) = (x2 – 2)(x3 – 27) = (x2 – 2)(x – 3)(x2 + 3x + 9)

Examples 4: a2 – b2 = (a + b)(a – b) a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 Factor a)100x2 – 60xy + 9y2 b) x3 – 1 8 y3 c) 3x2 – 18x + 27 – 3y2