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6.5 Factoring Cubic Polynomials

6.5 Factoring Cubic Polynomials. 2/13/2013. Perfect Cubes. 1000 = 10 3 729 = 9 3 512 = 8 3 343 = 7 3 216 = 6 3. 125 = 5 3 64 = 4 3 27 = 3 3 8 = 2 3 1 = 1 3. T he sum of two cubes :. The difference of two cubes :. Example 1. a. Factor . b.

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6.5 Factoring Cubic Polynomials

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  1. 6.5Factoring Cubic Polynomials 2/13/2013

  2. Perfect Cubes 1000= 103 729 = 93 512= 83 343= 73 216= 63 125 = 53 64= 43 27= 33 8= 23 1 = 13

  3. The sum of two cubes: The difference of two cubes:

  4. Example 1 a. Factor . b. Factor . x3 + 64 SOLUTION a. – x3 + 64 x3 + 43 8p3 q3 Write as sum of two cubes. = = Use special product pattern. = ( ( ) ) x x 4 4 + + Simplify. ( ( ) ) – – x2 x2 4x 4x + + 16 42 Factor the Sum or Difference of Two Cubes

  5. Example 1 Simplify. b. ( ) – – 2p 3 8p3 q3 q3 = [ ] ( ) ( ) – Use special product pattern. 2p 2 + 2pq + 2p q q2 = ( ) ( ) – + 2pq + 2p q 4p2 q2 = Factor the Sum or Difference of Two Cubes Write as difference of two cubes.

  6. Checkpoint ANSWER 1. x3 + 1 ( ) ( ) – x 1 + x2 x + 1 2. 125x3 + 8 ( ) ( ) – 5x 2 + 25x2 10x + 4 3. – x3 216 ( ) ( ) – x 6 x2 + 6x + 36 Factor the polynomial.

  7. Finding Greatest Common Factor (GCF) 3 2x Find the GCF of the terms in the polynomial:

  8. Example 2 – – x3 16x4 5x2 6x. 2x. + SOLUTION a. ( ) – – Factor common monomial. x3 x2 5x2 6x x 5x 6 + + = ( ) ( ) – – Factor trinomial. x 3 x 2 x = – 16x4 2x b. ( ) – Factor common monomial. 8x3 2x 1 4x2 2x 1 + + = ( ) ( ) – Use special product pattern. 2x 2x 1 = Factor Polynomials a. Factor b. Factor

  9. Checkpoint ANSWER ( ) ( ) – x 1 x + 3 x – 2x3 10x 2 8x + 5. ( ) ( ) – – x 4 x 1 2x – 54x4 16x 3x 4 24x + ( ) ( ) 6. x + 2 3x ( ) ( ) – 9x2 3x 2 6x 4 2x + + 7. – x2 2x 4 + Factor Polynomials Factor the polynomial. 4. – x3 2x 2 3x +

  10. Homework: WS 6.5 #1-18 skip #8

  11. Example 4 SOLUTION a. ( ) ( ) ( ) ( ) – – – – – x2 Use distributive property. x 1 9 x 1 x2 9 x 1 = ( ) ( ) ( ) – – Difference of two squares x 3 x + 3 x 1 = Factor by Grouping Factor the polynomial. a. b. ( ) ( ) – – – – – x3 x 1 9 x 1 x2 2x2 16x 32 +

  12. Example 4 ( ) ( ) – – – – 16x 32 2x2 16x 32 x3 2x2 x3 + + + = ( ) ( ) ( ) – – – Factor each group. 16 x2 x 2 x 2 + = ( ) ( ) Use distributive property. – – x2 16 x 2 = ( ) ( ) ( ) – – Difference of two squares x 4 x + 4 x 2 = Factor by Grouping b. Group terms.

  13. Checkpoint ANSWERS ( ) ( ) – x + 6 4 x + 6 x2 ( ) ( ) ( ) – x + 6 x 2 x + 2 9. – – 4x2 25x 100 x3 + ( ) ( ) ( ) – – x 4 x 5 x + 5 10. 3x2 4x 12 x3 + + + ( ) ( ) x + 3 x2 + 4 Factor by Grouping Factor the polynomial by grouping. 8.

  14. Example 5 – – 14x2 24x. 2x3 = – Rewrite in standard form. 14x2 24x 0 2x3 + = ( ) – Factor common monomial. 2x 7x 12 0 x2 + = ( ) ( ) – – Factor trinomial. x 4 x 3 0 2x = – – Use zero product property. or or 2x 0 x 4 0 x 3 0 = = = Solve for x. 0, 4, 3 x x x = = = Solve a Cubic Equation by Factoring Solve SOLUTION

  15. Example 6 – 6x2 12 2x. x3 + = – – 6x2 2x 12 0 x3 + = Rewrite in standard form. ( ) ( ) – – Group terms. 6x2 2x 12 0 x3 + + = ( ) ( ) ( ) – – – Factor each group. x 6 2 x 6 0 x2 + = ( ) ( ) – – Use distributive property. 2 x 6 0 x2 = – – Use zero product property. 2 0 or x 6 0 x2 = = Solve a Cubic Equation by Factoring Solve SOLUTION +2 +2 +6 +6 x = 6 x2 = 2

  16. Checkpoint ANSWER – 4, 0, 1 + + + – – – 14. – 3x3 30x 9x 13, 0 = 15. – – 2x2 3x 6 2, 3 x3 + = 16. – – 7x2 5x 35 x3 5, 7 = Solve a Cubic Equation by Factoring Solve the equation by factoring. 13. 3x2 4x x3 + =

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