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7.1 FACTORING POLYNOMIALS The greatest common factor (GCF) of an expression consists of the largest monomial that divides (is common to) all terms and any variables that appear in every term with an exponent equal to the smallest exponent of that variable in the expression. Factor out common factors: For numbers, look for a positive or negative number that divides evenly into all terms; for variables, if the same variable appears in all terms, factor out that variable using the smallest exponent. 1. 4a2b2 + 12ab3 + 4ab2 2. -13y8 + 26y4 – 39y2 3. 5(x + y) + a(x + y) 4. 2x2(x + 2) – 5x (x + 2) + 9(x + 2) Factor by grouping: If polynomial has four or more terms, separate it into two or more groups, factor any common factor out of each group and look for a common binomial factor. Remember that 1 and -1 are factors of anything. 5. 4x3 + 3x2y + 4xy2 + 3y3 6. 18r2 + 12ry – 3ry – 2y2 To factor a trinomial, x2 + bx + c, find factors of c whose sum = b. Check first for common factors; then factor by trial & error. When constant term c is positive, signs in the binomials will be the same and will match the middle term. When constant term c is negative, signs in the binomials will be different. 1. a2 + 9a + 20 2. b2 - 8b + 15 Look for factors of –42 whose sum is -1. Look for factors of 20 whose sum is 9. 4. n2 – 2n – 35 3. q2 – q – 42 5. 3y3z + 9y2z2 – 162yz3 6. 2m3n – 20m2n2 + 24mn3 To factor a trinomial, ax2 + bx + c, find factors of a and c where the sum of the inner and outer products = b; e.g., (a1 + c1)(a2 + c2) where a1c2+a2c1 = b. Check to see that you have factored completely. 7. 4x2 + 11x + 6 10. 3a2 + 6a + 3 8. 30a2 – 38a + 12 11. x3 – 5x2 – 6x 9. 7u2 + 11u - 6 12. 3x2(x + 5) – 19x(x + 5) – 14(x + 5) Special Factoring Differences of two squares: a2 – b2 = (a + b)(a – b) 1. 4x2 – 9 2. 9r2 – 1 3. 16y2 + 25 4. 100b2 – 4/49 5. 36m2 – 16/25 6. 32a3 – 8ab2 7. 36x5 – 16xy2 8. 16k4 – 1 9. y4 - 16 Perfect Square Trinomials: a2 + 2ab + b2 = (a + b)2 and a2 – 2ab + b2 = (a – b)2 10. 9x2 – 42xy + 49y2 11. 16x2 + 40x + 25 12. 3y2 – 48y + 192 13. 2x2 + 24x + 72 14. -18x2 – 48xy – 32y2 15. -50h2 + 40hy – 8y2 Difference of two cubes: a3 – b3 = (a – b)(a2 + ab + b2) Sum of two cubes: a3 + b3 = (a + b)(a2 – ab + b2) 16. m3 + 8 17. a3 - 1 18. 64x3 – 125y3 20. 3x4y – 15x2y – 108y 19. 27a3 + 343b3