Section 9-9Advanced Factoring Techniques

Warm-upFactor.

1. 2x3 + 3x2 + x 2. 2x3 + 2x2 + x2 + x

Warm-upFactor.

1. 2x3 + 3x2 + x 2. 2x3 + 2x2 + x2 + x

x(2x2 + 3x + 1)

Warm-upFactor.

1. 2x3 + 3x2 + x 2. 2x3 + 2x2 + x2 + x

x(2x2 + 3x + 1)

x(2x + 1)( x + 1)

Warm-upFactor.

1. 2x3 + 3x2 + x 2. 2x3 + 2x2 + x2 + x

x(2x2 + 3x + 1)

x(2x + 1)( x + 1)

(2x3 + 2x2 ) + ( x2 + x )

Warm-upFactor.

1. 2x3 + 3x2 + x 2. 2x3 + 2x2 + x2 + x

x(2x2 + 3x + 1)

x(2x + 1)( x + 1)

(2x3 + 2x2 ) + ( x2 + x )

2x2 ( x + 1) + x( x + 1)

Warm-upFactor.

1. 2x3 + 3x2 + x 2. 2x3 + 2x2 + x2 + x

x(2x2 + 3x + 1)

x(2x + 1)( x + 1)

(2x3 + 2x2 ) + ( x2 + x )

2x2 ( x + 1) + x( x + 1)

(2x2 + x )( x + 1)

Warm-upFactor.

1. 2x3 + 3x2 + x 2. 2x3 + 2x2 + x2 + x

x(2x2 + 3x + 1)

x(2x + 1)( x + 1)

(2x3 + 2x2 ) + ( x2 + x )

2x2 ( x + 1) + x( x + 1)

(2x2 + x )( x + 1)

x(2x + 1)( x + 1)

Factoring by Grouping

Factoring by Grouping

Grouping terms with common factors, then factoring out the GCF and applying distribution

Factoring by Grouping

Grouping terms with common factors, then factoring out the GCF and applying distribution

...you know, like we do with factoring quadratics

Example 1Factor

6a2b + 2ac + 3ab2 + bc

Example 1Factor

6a2b + 2ac + 3ab2 + bc

(6a2b + 2ac) + (3ab2 + bc)

Example 1Factor

6a2b + 2ac + 3ab2 + bc

(6a2b + 2ac) + (3ab2 + bc)

2a(3ab + c) + b(3ab + c)

Example 1Factor

6a2b + 2ac + 3ab2 + bc

(6a2b + 2ac) + (3ab2 + bc)

2a(3ab + c) + b(3ab + c)

(3ab + c)(2a + b)

Example 2Factor

4x2 + 4x − 15

Example 2Factor

4x2 + 4x − 15

4x2 + 10x − 6x − 15

Example 2Factor

4x2 + 4x − 15

(4x2 + 10x ) + (−6x − 15) 4x2 + 10x − 6x − 15

Example 2Factor

4x2 + 4x − 15

2x(2x + 5) − 3(2x − 5) (4x2 + 10x ) + (−6x − 15)

4x2 + 10x − 6x − 15

Example 2Factor

4x2 + 4x − 15

2x(2x + 5) − 3(2x − 5) (4x2 + 10x ) + (−6x − 15)

4x2 + 10x − 6x − 15

(2x − 3)(2x + 5)

Factoring by Grouping and Graphing

Factoring by Grouping and Graphing

Factoring into linear factors and graphing to see possible solutions

Example 3Describe the set of ordered pairs (x, y) satisfying

x3 + 3x2 − yx − 3 y = 0

Example 3Describe the set of ordered pairs (x, y) satisfying

x3 + 3x2 − yx − 3 y = 0

( x3 + 3x2 ) + (− yx − 3 y) = 0

Example 3Describe the set of ordered pairs (x, y) satisfying

x3 + 3x2 − yx − 3 y = 0

( x3 + 3x2 ) + (− yx − 3 y) = 0

x2 ( x + 3) − y( x + 3) = 0

Example 3Describe the set of ordered pairs (x, y) satisfying

x3 + 3x2 − yx − 3 y = 0

( x3 + 3x2 ) + (− yx − 3 y) = 0

x2 ( x + 3) − y( x + 3) = 0

( x2 − y)( x + 3) = 0

Example 3Describe the set of ordered pairs (x, y) satisfying

x3 + 3x2 − yx − 3 y = 0

( x3 + 3x2 ) + (− yx − 3 y) = 0

x2 ( x + 3) − y( x + 3) = 0

( x2 − y)( x + 3) = 0

x2 − y = 0

Example 3Describe the set of ordered pairs (x, y) satisfying

x3 + 3x2 − yx − 3 y = 0

( x3 + 3x2 ) + (− yx − 3 y) = 0

x2 ( x + 3) − y( x + 3) = 0

( x2 − y)( x + 3) = 0

x2 − y = 0 x + 3 = 0

Example 3Describe the set of ordered pairs (x, y) satisfying

x3 + 3x2 − yx − 3 y = 0

( x3 + 3x2 ) + (− yx − 3 y) = 0

x2 ( x + 3) − y( x + 3) = 0

( x2 − y)( x + 3) = 0

x2 − y = 0 x + 3 = 0

y = x2

Example 3Describe the set of ordered pairs (x, y) satisfying

x3 + 3x2 − yx − 3 y = 0

( x3 + 3x2 ) + (− yx − 3 y) = 0

x2 ( x + 3) − y( x + 3) = 0

( x2 − y)( x + 3) = 0

x2 − y = 0 x + 3 = 0

y = x2 x = 3

Example 3Describe the set of ordered pairs (x, y) satisfying

x3 + 3x2 − yx − 3 y = 0

( x3 + 3x2 ) + (− yx − 3 y) = 0

x2 ( x + 3) − y( x + 3) = 0

( x2 − y)( x + 3) = 0

x2 − y = 0 x + 3 = 0

y = x2 x = 3

The union of the sets of ordered pairs satisfying y = x2 and x = 3.

Homework

Homework

p. 610 #1-17