Chapter 5 Polynomials and Factoring

1
Chapter 5 - Polynomials and Factoring

• Section 1 Introduction to Factoring

2
Introduction

• Factor
• 18
• x2 7x

2?9
x(x 7)
x(x) 7(x)
3
Factor Monomials

• Factor 24y4

b)
a)
(6?4)(y3 ? y)
(8?3)(y2 ? y2)
(6 y3)(4 y)
(8y2)(3 y2)
4
Removing Common Factors

• Multiply
• x(x 9)

• Factoring

x2 9x
x?x x?9
x?x x?9
x2 9x

• x(x 9)

5
Removing Common Factors

• Multiply
• 8x(3x2 4x 9)

• Factor

24x3 32x2 72x
8x?3x2 8x?(-4x) 8x ?(-9)
8x?3x2 8x?(-4x) 8x ?(-9)
24x3 - 32x2 - 72x

• 8x(3x2 4x 9)

6
Factoring 1 of 4

• Factor
• 6x - 24

6?x - 6?4
(x 4)
6
7
Factoring 2 of 4

• Factor
• 8x2 32x

Largest Common Factor
x
8
8x?x 8x?4
(x 4)
8x
8
Factoring 3 of 4

• Factor
• 18x5 27x2

Largest Common Factor
x2
9
9x2?x3 9x2 ?3
(x3 3)
9x2
9
Factoring 4 of 4

• Factor
• 12x5 - 15x4 27x3

Largest Common Factor
x3
3
3x3?4x2 3 x3?5x 3 x3?9
(4x2 5x 9)
3x3
10
Practice

• Factor

Largest Common Factor
8
rs2
8rs2?r2 8rs2?2s
8rs2(r2 2s)
Largest Common Factor
-3
x
-3x?y (-3x)?(-2z) (-3x) ?1
-3x(y - 2z 1)
11
Factor by Grouping

• Factor

y3(y 2) 9(y 2)
(y 2)
(y3 9)
12
Factor by Grouping 1 of 3

• Factor

13
Factor by Grouping 2 of 3

• Factor

14
Factor by Grouping 3 of 3

• Factor

15
Practice

• Factor

16
Practice

• MathXL Ch 5 Section 1 Homework

17
Ch 5

• Section 2 Factoring Trinomials of the Type x2
  bx c

18
Introduction

• It is possible to factor polynomials like these
• x2 5x 4
• x2 3x -10

1

• Trinomials
• Coefficient of the Leading Term is 1.

1
19
FOIL Reviewed
First
Inner
Last
Outer

• (x 7)(x 1)

x2
7x
1x
7?1
x2 8x 7
20
FOIL Reviewed
First
Inner
Last
Outer

• (x 7)(x 1)

x2
7x
1x
7?1
x2 8x 7
71
21
FOIL Reviewed
First
Inner
Last
Outer

• (x 7)(x 1)

x2
7x
1x
7?1
x2 8x 7
7?1
22
One More
First
Inner
Last
Outer

• (x 3)(x 5)

x2
5x
3x
3?( 5)
x2 2x 15
23
One More
First
Inner
Last
Outer

• (x 3)(x 5)

x2
5x
3x
3?( 5)
x2 2x 15
3 5
24
One More
First
Inner
Last
Outer

• (x 3)(x 5)

x2
5x
3x
3?( 5)
x2 2x 15
3( 5)
25
Factoring Establishing a Process

• Factor

10
x2 7x 10
7
10?110
10111
( )( )
x x
a
b
5?210
527
(x 5)(x 2)
Is this correct?
x2 2x 5x 10
x2 7x 10
26
Factoring Example 1

• Factor

6
x2 5x 6
5
6?16
617
( )( )
x x
a
b
3?26
325
(x 3)(x 2)
Is this correct?
x2 2x 3x 6
x2 5x 6
27
Factoring Example 2

• Factor

12
y2 8y 12
8
-1(-12)12
-1(-12)-13
( )( )
y y
a
b
-2(-6)12
-2(-6) -8
(x 2)(x 6)
Is this correct?
x2 6x 2x 12
x2 8x 12
28
Factoring Example 3

• Factor

20
x2 8x 20
8
1?(-20) -20
1(-20) -19
( )( )
x x
a
b
2(-10) -8
2?(-10) -20
4?(-5) -20
4(-5) -1
5(-4) 1
5?(-4) -20
(x 2)(x 10)
10?(-2) -20
10(-2) 8
Is this correct?
20?(-1) -20
20(-1) 19
x2 10x 2x 20
x2 8x 20
29
Factoring Example 4

• Factor

24
t2 24 5t
t2 5t 24
5
-1?24 -24
-124 23
( )( )
t t
a
b
-2?12 -24
-212 10
-38 5
-3?8 -24
(t 8)(t 3)
-4?6 -24
-46 2
Is this correct?
t2 3t 8t 24
t2 5t 24
30
Factoring Example 5

• Factor

a2 4ba 21b2
a2 4ab 21b2
21b2
4b
-7b(3b) -21b2
-7b 3b -4b
( )( )
a a
r
s
7b(-3b) -21b2
7b (-3b) 4b
(a 3b)(a 7b)
Is this correct?
a2 7ab 3ab 21 b2
a2 4ab 21 b2
31
Factoring Example 6

• Factor

5
x2 x 5
1
1?5 5
15 6
( )( )
x x
a
b
-1?(-5) 5
-1(-5) -6
(x )(x )
Cannot Factor Prime Polynomial
32
2x3 20x2 50x
Factoring Example 7

• Factor

2x3 20x2 50x
10
( 2x)?x2 ( 2x)?(10x) ( 2x)25
2x(x2 10x 25)
25
-25?(-1) -25
-25 (-1) -26
-2x( )( )
x x
a
b
-5?(-5) 25
-5 (-5) -10
-2x(x 5)(x 5)
Is this correct?
x2 5x 5x 25
x2 10x 25
2x(x2 10x 25)
2x3 20x2 50x
33
Practice

• MathXL Ch 5 Section 2 Homework