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Mastering Factoring in Algebra

This chapter covers prime factoring, greatest common factor, factorization through the distributive property, grouping, zero product property, and factoring trinomials and differences of squares in algebraic expressions.

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Mastering Factoring in Algebra

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  1. Chapter 8: Factoring

  2. Fill in the titles on the foldable

  3. 8.1 Prime factoring and factor a monomial (top) • Prime # = factors only include 1 and itself • Composite # = more than two factors Ex: Prime factor 90 Prime numbers: 1, 3, 5, 7, 11, 13, 17, 23, 29, 31, 37 …. 90 2 45 3 15 3 5 2 x 3 x 3 x 5 = 2 x 32 x 5

  4. 8.1 Prime factoring and factor a monomial (bottom) • Factor a monomial = list all factors separately (no exponents) Ex: -66pq2 Ex: 38rs2t -66 38 66 -1 2 19 2 33 2 x 19 x r x s x s x t 3 11 -1 x 2 x 3 x 11 x p x q x q

  5. 8.1 Greatest Common Factor (top) • GCF = the largest factor that is in all the given monomials • 1. factor all monomials • 2. circle all common factors • 3. Multiply all the circled factors

  6. 8.1 Greatest Common Factor (bottom) Ex: 84 & 70 Ex: 36x2y & 54xy2z 36 84 70 54 2 18 2 42 2 35 2 27 2 9 2 21 5 7 3 9 3 3 3 7 3 3 2 x 2 x 3 x 7 2 x 2 x 3 x 3 x x x x x y 2 x 5 x 7 2 x 3 x 3 x 3 x x x y x y x z 2 x 7= 14 2 x 3 x 3 x x x y= 18xy

  7. 8.2 Factor Using Distributive Property (top) • Find the GCF of the monomials • Write each term as a product of the GCF and the remaining factors • Simplify the remaining factors

  8. 8.2 Factor Using Distributive Property (bottom) Ex: 12a2 + 16a Ex: 3p2q – 9pq2 + 36pq -9 36 12 16 -1 9 2 18 2 6 2 8 3 3 2 9 2 3 2 4 3 3 2 2 x 3 x p x p x q 2 x 2 x 3 x a x a -1 x 3 x 3 x p x q x q = 2 x 2 x a =4a 3 x p x q = 3pq 2 x 2 x 2 x 2 x a 2 x 2 x 3 x 3 x p x q 4a(3a) + 4a(4) = 3pq(p) + 3pq(-3q) + 3pq(12) = 4a(3a + 4) 3pq(p - 3q + 12)

  9. 8.2 Factor by Grouping (top) • Group the terms (first two and last two) • Find the GCF of each group • Write each group as a product of the GCF and the remaining factors • Combine the GCFs in a group and write the other group as the second factor

  10. 8.2 Factor by Grouping (bottom) Ex: 4ab + 8b + 3a + 6 Ex: 3p – 2p2 – 18p + 27 (4ab + 8b)(+ 3a + 6) (3p – 2p2 )( – 18p + 27) -18 4 6 27 8 -1 18 2 2 2 3 3 9 2 4 2 9 3 3 2 2 3 3 2 x 2 x a x b 3 x a 3 x p = 3 = p -1 x 2 x 3 x 3 x p =4b 2 x 2 x 2 x b 2 x 3 -1 2 x p x p = 9 3 x 3 x 3 4b(a + 2) +3 (a + 2) p(3 – 2p) + 9(-2p + 3) (4b + 3)(a + 2) (p + 9)(-2p + 3)

  11. 8.2 Zero Product Property (top) • Roots = the solutions to the equation • When an equation is factored and equal to zero: Set each factor equal to zero and solve for the variable

  12. 8.2 Zero Product Property (bottom) Ex: 7f2 – 35f = 0 Ex: (d – 5)(3d + 4) = 0 -35 d – 5 = 0 3d + 4 = 0 7 x f x f = 7f + 5 + 5 - 4 - 4 -1 x 5 x 7 x f -1 35 3d = -4 d = 5 5 7 /3 /3 d = -4/3 7f(f) + 7f(-5) 7f(f – 5) = 0 Roots are d = 5 and -4/3 f – 5 = 0 7f = 0 + 5 + 5 /7 /7 f = 5 f = 0 Roots are f = 0 and 5

  13. 8.3 Factoring Trinomials – x2 + bx + c (top) • Get everything on one side (equal to zero) • Split into two groups ( )( ) = 0 • Factor the first part x2 (x )(x ) = 0 • Find all the factors of the third part (part c) • Fill in the factors of c that will add or subtract to make the second part (bx) • Foil to check your answer • Use Zero Product Property to solve if needed

  14. 8.3 Factoring Trinomials – x2 + bx + c (bottom) Ex: x2 + 6x + 8 Ex: r2 – 2r - 24 Ex: s2 – 11s + 28 = 0 8 1, 8 2, 4 (x )(x ) 24 1, 24 2, 12 3, 8 4, 6 28 1, 28 2, 14 4, 7 (r )(r ) (s )(s ) (x + 2)(x + 4) (s- 4)(s - 7) = 0 (r + 4)(r - 6) FOIL x2 + 2x + 4x + 8 x2 + 6x + 8 FOIL s2 – 7s – 4s + 28 s2 – 11s + 28 FOIL r2 – 6r + 4r - 24 r2 - 2x - 24 s – 4 = 0 s – 7 = 0 +4 +4 +7 +7 s = 4 s = 7 s = 4 and 7

  15. 8.4 Factoring Trinomials – ax2 + bx + c (top) • Get everything on one side (equal to zero) • Put the first part in each set of parentheses • Find product of the first and last parts • Find the factors of the product • Fill in the pair of factors that adds or subtracts to the second part • Remove the GCF from one set of parentheses • Write what is left of the that group as one factor and then the other group as the other factor • if you can’t factor = prime (use the zero product property to solve if needed)

  16. 8.4 Factoring Trinomials – ax2 + bx + c (bottom) Hint: find the gcf to pull it out and make the numbers smaller if possible Ex: 5x2 + 13x + 6 Ex: 10y2 - 35y + 30 = 0 2 x 6 = 12 5 x 6 = 30 5(2y2 - 7y + 6) = 0 1, 12 2, 6 3, 4 (5x )(5x ) 1, 30 2, 15 3, 10 5, 6 5(2y )(2y )=0 (5x + 10)(5x + 3) 5(2y - 4)(2y - 3)=0 5x: x 5 = 5 2 x y 10: 2 5 = 2 -1 x2 x 2 (5x + 10) 5(x + 2) (2y - 4) 2(y - 2) (x + 2)(5x + 3) 5(y - 2)(2y - 3) = 0 Solve for y. y – 2 = 0 2y – 3 = 0 y = 2 and 1.5

  17. 8.5 Factoring Differences of Squares (top) • Factor each term • Write one set of parentheses with the factors adding and one with the factors subtracting • Foil to check your answer Hint: find the gcf to pull it out and make the numbers smaller if possible Ex: n2 - 25 Ex: 9x3 – 4x n x n 5 x 5 x(9x2 – 4) (n + 5)(n - 5) x[ 3x x 3x 2 x 2] x(3x + 2)(3x - 2)

  18. 8.5 Factoring Differences of Squares (bottom) Ex: 5x3 + 15x2 – 5x - 15 Ex: 121a = 49a3 -121a -121a 5[x3 + 3x2 – x – 3] 0 = 49a3 – 121a 5[ (x3 + 3x2)( – x – 3)] 0 = a(49a2 – 121) 0 = a(7a x 7a 11 x 11) 0 = a(7a + 11)(7a - 11) x x x x x -1 x x = x2 = -1 3 x x x x -1 x 3 a = 0 7a + 11 = 0 7a - 11 = 0 -11 -11 +11 +11 5[ x2(x + 3) - 1(x + 3)] 7a = -11 7a = 11 /7 /7 /7 /7 5[(x2 – 1)(x + 3)] 5[(x x x 1 x 1)(x + 3)] a= -11/7 a = 11/7 5(x + 1)(x - 1)(x + 3) a = -11/7, 0, and 11/7

  19. 8.6 Factoring Perfect Squares (top) • Perfect Square Trinomial: • Is the first term a perfect square? • Is the last term a perfect square (must be a positive number)? • Does the second term = 2 x the product of the roots of the first and last terms? • If any of these answers is no- it is not a perfect square trinomial

  20. 8.6 Factoring Perfect Squares (bottom) Ex: x2 – 14x + 49 Ex: a2 – 8a - 16 x x x 7 x 7 a x a 4 x 4 2 x x x 7= 14x 4 x 4 = 16 but it is a negative 16 so it can’t be a perfect square (x – 7)2 Ex: 9y2 + 12y + 4 1. 9y2 = 3y x 3y yes 2. 4 = 2 x 2 yes 3. 2(3y x 2) = 2(6y) = 12y yes (3y + 2)2

  21. 8.7 Square Root Property Ex: (y – 8)2 = 7 Ex: (b – 7)2 = 36 b – 7 = 6 b – 7 = -6 +8 +8 +8 +8 +7 +7 +7 +7 b = 13 b = 1

  22. 11.2 Rational Expressions(top) • You can only cancel factors if they are exactly the same groups Ex:

  23. 11.2 Rational Expressions(bottom) Ex:

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