1. 6.5 Factoring Cubic Polynomials 1/31/2014

2. Cube: a geometric figure where all sides are equal. 10 in Volume of a cube: side sideside V= 10 1010 V = 1000 in 3

3. 1000 = 10 3 729 = 9 3 512 = 8 3 343 = 7 3 216 = 6 3 Perfect Cubes (Volume of a cube whose sides are whole numbers) 125 = 5 3 64 = 4 3 27 = 3 3 8 = 2 3 1 = 1 3 The side length is the CUBE ROOT of the perfect cube.

4. Multiply the following:

5. Summary: In Reverse: If you were asked to factor: Factor of the sum of two cubes: Square the 1st term base Multiply the first and second base Square the 2nd term base First term base 2nd term base

6. Factor of the difference of two cubes: Factor:

7. Example 1 Factor the Sum or Difference of Two Cubes a.Factor. x 3x 3 + 216 b.Factor. 8p 38p 3 – q 3q 3 SOLUTION Write as sum of two cubes. x 3x 3 +216 = x 3x 3 +6363 a. () 6x + () x 2x 2 6x6x+ – 6262 = () 6x + () x 2x 2 6x6x+ – 36 =

8. Example 1 Factor the Sum or Difference of Two Cubes = – () q2p2p + q2q2 2pq+ 4p24p2 () b. 8p 38p 3 – q 3q 3 – () 2p 3 q 3q 3 = Write as difference of two cubes. = – () q2p2p + q2q2 2pq2pq [] () 2p 2 +

9. Checkpoint Factor the polynomial. 1. x 3x 3 +1 2. 125x 3 +8 ANSWER () 1x + () x 2x 2 x+ – 1 () 25x5x + () 25x 2 10x+ – 4 3. x 3x 3 216 – () 6x + () x 2x 2 6x6x+36 –

10. Finding Greatest Common Factor (GCF)

11. Example 2 Factor Polynomials completely a. Factor x 3x 3 5x 25x 2 6x.6x. + – b. Factor 16x 4 2x.2x. – = x () 3x – () 2x – SOLUTION x 3x 3 5x 25x 2 6x6x + – = a. = b. 16x 4 2x2x – = () 2x2x 2x2x 1 – 4x 24x 2 2x2x1 ++ () Factor using Big X GCF: x GCF: 2 x Factor out the GCF: x Factor out the GCF: 2 x

12. Checkpoint Factor the polynomial. x 3x 3 2x 2 3x3x + – 4. Factor Polynomials 5. 2x 32x 3 10x 2 8x8x – + ANSWER x () 1x – () 3x+ 2x2x () 4x – () 1x –

13. Checkpoint Factor the polynomial. Factor Polynomials 6. 3x 4 24x + 7. 54x 4 16x – 3x3x () 2x+ () x 2x 2 2x2x4 + – 2x2x () 23x3x () 9x 29x 2 6x6x4 + – + ANSWER

14. Homework: 6.5 p.326 #3-6, 14-19, 28-32

15. 6.5 Factoring Cubic Polynomials…. cont’d 2/3/2014

16. Example 4 Factor by Grouping Factor the polynomial. a. x 2x 2 () 1x – () 1x – 9 – SOLUTION Factor our (x-1). a. x 2x 2 () 1x – () 1x – 9 – = () 9x 2x 2 – () 1x – = () 3x – () 3x+ () 1x – a 2 – b 2 pattern

17. Example 4 Factor by Grouping Factor each group using GCF. = ) x 2x 2 – ( – 2 + 16 ( x )) – 2 ( x Factor our (x – 2). = ) – 16 () – 2 ( xx 2x 2 a 2 – b 2 pattern = () 4x – () 4x+ () 2x – Group terms. = () x 3x 3 – () 32 – 2x 22x 2 + 16x + b. x 3x 3 2x 22x 2 16x –– 32 +

18. Checkpoint Factor the polynomial by grouping. 8. Factor by Grouping x 2x 2 () 6x+ () 6x+4 – 9. x 3x 3 4x 24x 2 25x –– 100 + 10. x 3x 3 3x 23x 2 4x4x12 +++ ANSWERS () 2x – () 2x+ () 6x+ () 5x – () 5x+ () 4x – () 3x+ () 4x 2x 2 +

19. Example 5 Solve a Cubic Equation by Factoring Solve 2x 32x 3 14x 2 – = 24x. – 2x 32x 3 14x 2 – = 24x + Rewrite in standard form. 0 () x 2x 2 7x7x – = 12 + Factor common monomial. 02x2x () 4x – () 3x – 2x2x = Factor trinomial. 0 Use zero product property. 4x – 3x – 2x2x = 0 = 0 or = 0 x = 0,0, x = 4,4, x = 3 Solve for x. SOLUTION

20. Example 6 Solve a Cubic Equation by Factoring Solve x 3x 3 6x 26x 2 – = 2x.2x. + 12 Rewrite in standard form. x 3x 3 6x 26x 2 – = 0 + 122x2x – () x 3x 3 6x 26x 2 – = + Group terms. 0 ) + ( 2x2x – () 6x – () 6x – x 2x 2 = Factor each group. 0 + )( 2 – () 6x – = Use distributive property. 0 () 2 – x 2x 2 6x – = Use zero product property. 02 – x 2x 2 = 0 or SOLUTION +2 +6 x 2 = 2 x = 6

21. Checkpoint Solve the equation by factoring. Solve a Cubic Equation by Factoring 13. x 3x 3 3x 23x 2 4x4x + = 14. 3x 33x 3 30x – = 9x9x 15. x 3x 3 2x 22x 2 3x3x + – 6 = 16. x 3x 3 7x 27x 2 5x5x = – 35 – ANSWER 4, 0, 1 – 13, 0 + – 2,2, – 3 + – 5, 7 + –

22. Homework: WS 6.5 Prac A