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Factoring

Understand various polynomial factoring techniques such as Greatest Common Factor, Grouping, and Reverse Foil. Practice with examples and shortcuts to simplify complex expressions efficiently.

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Factoring

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  1. Factoring Greatest Common Factor 2 5x- 15x = 5x(1-3x) 2 4 3 Grouping 5x – 10x – 2x + 4 5x (x – 2) – 2(x - 2) (5x – 2) (x – 2) Difference of Squares a - b = (a – b) (a+b) 4x – 49 = (2x – 7)(2x + 7) Difference of Cubes a + b = (a + b) (a + ab + b ) 8x – 1 = (2x – 1)(4x + 2x + 1) Reverse Foil Try to find the middle through trial and error Ax + Bx + c or Ax + Bx + c ( + + ) = ( + ) ( + ) ( - + ) = ( - ) ( - ) ( + - ) = ( - )( + ) or (+)(-) ( - - ) = ( - )( + ) or (+)(-) 3 2 2 2 2 2 2 4 2 2 2 - 2 3 3 3 2

  2. Factoring Polynomials • Different Ways to Factor • GCF (Greatest Common Factor) • Grouping • Reverse Foil • Calculator with Grouping • Cube Pattern

  3. GCF • Factor 4x²y + 8x³y. • What is the largest thing that both terms have in common? • What should we do when we find the GCF? • Write it in its factored form.

  4. Factor the following • 20a²b³ + 30a5b² • 33p4r³ - 9pr³ • 45xyz³ - 36x³yz + 18xy³z

  5. Grouping • A lot like GCF, just group terms first. • 4xy + 2x + 6y + 3 • 21 – 7t + 3r – rt • 4x² - 4xy + 8x – 8y

  6. Reverse Foil • Works with TRInomials • Always look for a GCF first!!! • x² + 5x + 6 • Using what you already know about FOIL, try to come up with a basic idea of what the factored form would look like.

  7. Factor Each Expression • x² + 7x + 6 • x² - 6x + 8 • x² + 11x + 24 • x² - 7x – 18 • x² + 3x – 10 • x² - 4x – 12 • 2x² + 3x + 1

  8. Special case in factoring… • Factor x² - 9 • How could we re-write x² - 9? • Now factor. • x² - 100 • 4x² - 49

  9. Now for some fun ones… • 3x² - 27y² • p4 – 1 • y4 – 81 • 3x² - 3y²

  10. What about polynomials with leading coefficients? • Factor 2x² + 5x + 3. • Here’s a shortcut… • a●c (2 ●3=6) • Rewrite as x² + 5x + 6, then factor. • (x + )(x + )

  11. Then divide out the 2 we originally multiplied (it was the coefficient of the x²). • (x + 3/2)(x + 2/2) • “Reduce” the fractions and if there is still a denominator, put it in the front.

  12. Factor the following polynomials. • 5x² + 34x + 24 • 6x² + x – 15 • 15x² + 26x + 8 • 4x² - 22x + 30 (hint: find a GCF first)

  13. Factor 8x³ - 1 • Take the cube root of each term • Cube root of 8x³ = 2x • Cube root of -1 = -1 • (2x – 1) • Square each of those and put them in a trinomial at the beginning and end • (4x + 1)

  14. (2x – 1)(4x + 1) • Multiply the terms from the binomial and change the sign • 2x●-1 = -2x 2x • Put that in the middle of the trinomial • (2x – 1)(4x + 2x + 1)

  15. Factor the following • 8x³ + 27 • x³ - 64 • x³ + 8y³ • -27x³ + 8

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