Polynomials: Factoring - XYZ Custom Plus

Chapter 12 SummaryGreatest Common Factor [12.1]The largest monomial that divides each term of a polynomial is called the greatestcommon factor for that polynomial. We begin all factoring by factoring out thegreatest common factor.EXAMPLEs1. 8x 4 − 10x 3 + 6x 2= 2x 2 ⋅ 4x 2 − 2x 2 ⋅ 5x + 2x 2 ⋅ 3= 2x 2 (4x 2 − 5x + 3)Factoring Trinomials [12.2, 12.3]One method of factoring a trinomial is to list all pairs of binomials the product ofwhose first terms gives the first term of the trinomial and the product of whoselast terms gives the last term of the trinomial. We then choose the pair that givesthe correct middle term for the original trinomial.2. x 2 + 5x + 6 = (x + 2)(x + 3)x 2 − 5x + 6 = (x − 2)(x − 3)6x 2 − x − 2 = (2x + 1)(3x − 2)6x 2 + 7x + 2 = (2x + 1)(3x + 2)Special Factorings [12.4]a 2 + 2ab + b 2 = (a + b) 2a 2 − 2ab + b 2 = (a − b) 23. x 2 + 10x + 25 = (x + 5) 2x 2 − 10x + 25 = (x − 5) 2x 2 − 25 = (x + 5)(x − 5)a 2 − b 2 = (a + b)(a − b)Sum and Difference of Two Cubes [12.5]a 3 − b 3 = (a − b)(a 2 + ab + b 2 )a 3 + b 3 = (a + b)(a 2 − ab + b 2 )Difference of two cubesSum of two cubes4. x 3 − 27 = (x − 3)(x 2 + 3x + 9)x 3 + 27 = (x + 3)(x 2 − 3x + 9)Strategy for Factoring a Polynomial [12.6]Step 1: If the polynomial has a greatest common factor other than 1, then factorout the greatest common factor.Step 2: If the polynomial has two terms (it is a binomial), then see if it is the differenceof two squares or the sum or difference of two cubes, and thenfactor accordingly. Remember, if it is the sum of two squares, it will notfactor.Step 3: If the polynomial has three terms (a trinomial), then it is either a perfectsquare trinomial that will factor into the square of a binomial, or it isnot a perfect square trinomial, in which case you use the trial and errormethod developed in Section 6.3.5. a. 2x 5 − 8x 3 = 2x 3 (x 2 − 4)= 2x 3 (x + 2)(x − 2)b. 3x 4 − 18x 3 + 27x 2= 3x 2 (x 2 − 6x + 9)= 3x 2 (x − 3) 2c. 6x 3 − 12x 2 − 48x= 6x (x 2 − 2x − 8)= 6x (x − 4)(x + 2)d. x 2 + ax + bx + ab= x (x + a) + b(x + a)= (x + a)(x + b)Step 4: If the polynomial has more than three terms, then try to factor it bygrouping.Step 5: As a final check, see if any of the factors you have written can be factoredfurther. If you have overlooked a common factor, you can catch ithere.Chapter 12Summary797