6.5Factoring Cubic Polynomials

1/31/2014

Cube: a geometric figure where all sides are equal.

10 in

10 in

10 in

Volume of a cube: side •side•sideV= 10 •10•10V = 1000 in3

1000 = 103

729 = 93

512 = 83

343 = 73

216 = 63

Perfect Cubes(Volume of a cube whose sides are

whole numbers)125 = 53

64 = 43

27 = 33

8 = 23

1 = 13

The side length is the CUBE ROOT of the perfect cube.

Multiply the following:𝑥2− 4 𝑥+16

𝑥+4

4 𝑥2 −16 𝑥+64𝑥3− 4 𝑥2+16 𝑥

𝑥3+64

4 𝑥2 −6 𝑥+9

2 𝑥+3

12𝑥2 −18 𝑥+278 𝑥3− 12𝑥2+18 𝑥

8 𝑥3+27(𝑥 )3+(4)3 (2 𝑥 )3+(3)3

Summary:(𝑥¿¿2 − 4 𝑥+16 )(𝑥+4)=𝑥3+64=   (𝑥 )3+(4 )3¿(4 𝑥¿¿2−6 𝑥+9)(2 𝑥+3)=8 𝑥3+27=   (2𝑥 )3+(3)3 ¿

In Reverse: If you were asked to factor: =   (𝑥 )3+(4 )3=(𝑥¿¿2− 4 𝑥+16)(𝑥+4)¿

= (2 𝑥 )3+(3)3=(4 𝑥¿¿2−6 𝑥+9)(2𝑥+3)¿Factor of the sum of two cubes: babababa 2233

Square the 1st term base

Multiply the first and second base

Square the 2nd term base

First term base

2nd term base

babababa 2233

Factor of the difference of two cubes:

Factor: =

=

  (𝑥 )3 − ( 4 )3=(𝑥¿¿2+4 𝑥+16)(𝑥− 4 )¿

(2 𝑥 )3 − (3 )3=(4 𝑥¿¿2+6 𝑥+9)(2 𝑥−3)¿

Example 1 Factor the Sum or Difference of Two Cubes

a. Factor .x 3 + 216 b. Factor .8p 3 – q 3

SOLUTION

Write as sum of two cubes.

x 3 + 216 = x 3 + 63a.

( )6x + ( )x 2 6x +– 62=

( )6x + ( )x 2 6x +– 36=

2233 babababa

Example 1 Factor the Sum or Difference of Two Cubes

= –( )q2p + q22pq+4p2( )

b. 8p 3 – q 3 –( )2p 3 q 3=Write as difference of two cubes.

= –( )q2p + q22pq[ ]( )2p 2 +

2233 babababa

Checkpoint Factor the polynomial.

1. x 3 + 1

2. 125x 3 + 8

ANSWER

( )1x + ( )x 2 x +– 1

( )25x + ( )25x 2 10x +– 4

3. x 3 216– ( )6x +( )x 2 6x + 36–

2233 babababa 2233 babababa

Finding Greatest Common Factor (GCF)

Find the GCF of the terms in the polynomial:

32x

Example 2 Factor Polynomials completely

a. Factor x 3 5x 2 6x.+– b. Factor 16x 4 2x.–

= x( )3x – ( )2x –

SOLUTION

x 3 5x 2 6x+– =a.

=b.

16x 4 2x–

= ( )2x 2x 1– 4x 2 2x 1+ +( )

2233 babababa

Factor using Big X

𝑥 (𝑥2 −5 𝑥+6)GCF: x

GCF: 2x2 𝑥(8𝑥3 −1)

Factor out the GCF: x

Factor out the GCF: 2x

Checkpoint

Factor the polynomial.

x 3 2x 2 3x+ –4.

Factor Polynomials

5. 2x 3 10x 2 8x– +

ANSWER

x( )1x – ( )3x +

2x( )4x – ( )1x –

Checkpoint

Factor the polynomial.

Factor Polynomials

6. 3x 4 24x+

7. 54x 4 16x–

3x( )2x + ( )x 2 2x 4+–

2x( )23x ( )9x 2 6x 4+– +

ANSWER

Homework:

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

6.5Factoring Cubic

Polynomials….cont’d

2/3/2014

Example 4 Factor by Grouping

Factor the polynomial.

a. x 2 ( )1x – ( )1x –9–

SOLUTION

Factor our (x-1).a. x 2 ( )1x – ( )1x –9– = ( )9x 2 – ( )1x –

= ( )3x – ( )3x + ( )1x – a2 – b2 pattern

Example 4 Factor by Grouping

Factor each group using GCF.

= )x 2 – ( –2 + 16( x ) )– 2( x

Factor our (x – 2).= )– 16( )– 2( xx 2

a2 – b2 pattern= ( )4x – ( )4x + ( )2x –

Group terms. = ( )x 3 – ( )32–2x 2 + 16x +

b. x 3 2x 2 16x– – 32+

Checkpoint

Factor the polynomial by grouping.

8.

Factor by Grouping

x 2 ( )6x + ( )6x +4–

9. x 3 4x 2 25x– – 100+

10.

x 3 3x 2 4x 12++ +

ANSWERS

( )2x – ( )2x +( )6x +

( )5x – ( )5x +( )4x –

( )3x + ( )4x 2 +

Example 5 Solve a Cubic Equation by Factoring

Solve 2x 3 14x 2– = 24x.–

2x 3 14x 2– =24x+ Rewrite in standard form.0

( )x 2 7x– =12+ Factor common monomial. 02x

( )4x – ( )3x –2x = Factor trinomial.0

Use zero product property.4x – 3x –2x = 0= 0or= 0 or

x = 0, x = 4, x = 3 Solve for x.

SOLUTION

Example 6 Solve a Cubic Equation by Factoring

Solve x 3 6x 2– = 2x.+ 12

Rewrite in standard form.x 3 6x 2– = 0+ 122x–

( )x 3 6x 2– =12+ Group terms.0) + ( 2x–

( )6x – ( )6x –x 2 = Factor each group.0+ )( 2–

( )6x – = Use distributive property.0( )2–x 2

6x – = Use zero product property.02–x 2 = 0 or

SOLUTION

+2 +2 +6 +6

x2 = 2 x = 6

Checkpoint

Solve the equation by factoring.

Solve a Cubic Equation by Factoring

13.

x 3 3x 2 4x+ =

14.

3x 3 30x– = 9x

15.

x 3 2x 2 3x+ – 6=

16.

x 3 7x 2 5x=– 35–

ANSWER

4, 0, 1–

13, 0+–

2, – 3+–

5, 7+–

Homework:

WS 6.5 Prac A