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Understanding Factoring: Methods and Examples for Polynomials

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This resource provides a comprehensive overview of factoring in algebra, focusing on polynomials. It covers fundamental concepts, including identifying squares and square roots, and introduces key operations such as addition and multiplication of polynomials. The document also explains different factoring techniques, including the Greatest Common Factor (GCF), Difference of Squares, and factoring by grouping. Through detailed examples and methods such as the Distributive Property and FOIL, learners can gain a strong foundation in solving polynomial equations and understanding their applications in mathematics.

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Understanding Factoring: Methods and Examples for Polynomials

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  1. FACTORING 11/29/2012

  2. Warm UpFind the square or square root of each. • 32 • 162 • (-12)2 • (2x)2 • (x+1)2 • (3x-4)2 7) √100 8) √225 9) √x2 10) √9x2 11) √(x-3)2 12) √x2 + 2x + 1

  3. Term 1 x x² 4x³ x²y³  Monomial Binomial x + 1 3x4y + y2z Trinomial 5x3 – x2 + 10 Polynomial 3xyz + 3xy2z - 0.1xz - 200y + 0.5

  4. Polynomial “poly” =many “-nomial” = terms

  5. Polynomials and Operations Addition + Multiplication  x  x 2x  x 3x  x 2x  3x 5x  9x x²  x² x²  x x + x 2x + x 3x + x 2x + 3x 5x + 9x x² + x² x² + x

  6. Polynomials and Operations Addition + Multiplication  (7x)(x²)(-3x) (7x²)(x²)(7x³) 7x + x² – 3x 7x² + x² + 7x³ What if? What if?

  7. Distributive Property To multiply a term outside the parenthesis with every term inside the parenthesis Examples: 5 (1 + 2) -3p(2p2 + 3pq – 5q) (x+2)(x-2) = 5(1) + 5(2) = 5 + 10 = 15 = -3x(2y2) + -3x(3xy) + -3x(-5y) = -6x3 – 9x2y + 15xy = x(x-2) + 2(x-2) =x2 -2x + 2x -4 = x2 - 4

  8. Distributing Methods • Box Method • (x-1)(x-2) • FOIL (x-1)(x-2) F O I L x2 + x - 2x + 2 x -1 X -2

  9. Distribution Multiplication  over Addition + = 1) 3(x + 1) 2) - (x + 1) 3) -2(x + 1) 4) x( x + 1) 5) (x + 1)(x + 1) 6) (x + 1)² 7) (x + 1)³

  10. ClassworkHandout

  11. What two numbers multiplied together equals… 12x 24x 9x² • 12 • 24 • 20

  12. Factor • To break, reduce, or take a term apart into smaller terms • To rewrite as a product of two or more terms • Opposite of distribute • There are 7 main forms of factoring, which you will learn today and next week.

  13. Factor the following -32x + 2x3 x2 – 11 x + 18 x2 + 11 x + 18 x2 – 2x – 15 x2 + 2x - 15 Ms. Constanza, I forgot and I don’t know how. Can you please TEACH ME HOW TO FACTOR http://www.youtube.com/watch?v=OFSrINhfNsQ

  14. The 7 Forms of factoring Always 2 Terms 3 Terms 4 Terms Greatest Common factor (G.C.F.) Difference of two Squares Difference of two Cubes Sum of Two Cubes Perfect Square Trinomial A Quadratic Trinomial Factor by Grouping

  15. GCF Complete the factorization for each • 3x² + 15x = 3( ) • 3x² + 15x = x( ) • 3x² + 15x = 3x( ) • Which of the above factorization for 3x² + 15x do you think is best? Why?

  16. FACTOR by finding the GCFif the terms do not have a GCF, then they are relatively prime • 5x + 5 • 3x² + x • 15x² + 5x • 2x² - 18 • 6x² - 3x + 9 • x³ - 5x • 7x - 9

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