1- Special Cases 3ol25
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Unit 3 Factoring Special Forms in order to find all zeroes of polynomials
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
REVIEW •When Factoring: Write the polynomial as a product. Steps: 1. Find the GCF of all its . 2. Write the polynomial as a product by factoring out the GCF from all the . - This is done by dividing the original of the polynomial by the GCF.
3. The remaining factors in each term will form a polynomial.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
The Difference of Two Squares
The Difference of Two Squares If A and B are real numbers, variables, or algebraic expressions, then
A2 B 2 A B A B . In words: The difference of the squares of two , factors as the product of a sum and a difference of those .
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
4
The Difference of Two Squares EXAMPLE
Factor: 25 x 4 9 y 6 . SOLUTION
We must express each term as the square of a monomial. Then use the formula for factoring A2 B 2 A B A B . 25 x 4 9 y 6
5x 3 y 2 2
5x
2
3 2
3 y3 5x2 3 y3
Express as the difference of two squares
Factor using the Difference of Two Squares method
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
The Difference of Two Squares EXAMPLE
Factor: 6 x 2 6 y 2. SOLUTION
The GCF of the two of the polynomial is 6. We begin by factoring out 6. 6x2 6 y2
6 x2 y 2
6 x y x y
Factor the GCF out of both Factor using the Difference of Two Squares method
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
The Difference of Two Squares EXAMPLE
Factor completely: x 4 1. SOLUTION
x 1 x 4
2 2
12
Express as the difference of two squares
x2 1 x2 1
x 2 1 x 2 12
The factors are the sum and difference of the expressions being squared
The factor x 2 1 is the difference of two squares and can be factored
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
The Difference of Two Squares CONTINUED
x 2 1 x 1 x 1
The factors of x 2 1 are the sum and difference of the expressions being squared
Thus, x 4 1 x 2 1 x 1 x 1 .
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Factoring Completely by Grouping EXAMPLE
Factor completely: x 3 3x 2 9 x 27. SOLUTION
x 3 3x 2 9 x 27
x 3 3 x 2 9 x 27
x 2 x 3 9 x 3
x 3 x 2 9
x 3 x 3 x 3
Group with common factors Factor out the common factor from each group Factor out x + 3 from both Factor x 2 9 as the difference of two squares
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example “ Difference of 2 Squares
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10
Objective #1: Example
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11
Objective #1: You Try! Example Difference of Two Squares
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Objective #1: Example
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
• Objective 2: Factoring Sum and Difference of 2 perfect Cubes
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
The Sum & Difference of Two Cubes Factoring the Sum & Difference of Two Cubes 1)
Factoring the Sum of Two Cubes
A3 B 3 A B A2 AB B 2 Same Signs
Opposite Signs
2) Factoring the Difference of Two Cubes
A3 B 3 A B A2 AB B 2 Same Signs
Opposite Signs
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Write each monomial as a cube and apply either of the rules.
Example :
Rewrite as cubes 3
3
x 64 (x 4 ) 3
Apply the rule for sum of 3 3 cubes: a b ab
a ab b
2
2
2
2
(x 4)(x x 4 4 ) 2
(x 4)(x 4x 16) Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Rewrite as cubes 3 3
3
Ex: 8y 125 ((2y) 5 ) Apply the rule for difference of cubes:
a b a ba ab b 3
3
2
2
2y 5 2y 2y 5 5 2
2
2y 54y 10y 25
2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #2: Example
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18
Objective #2 : Example
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19
Objective #2 : Example
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20
Objective #2 : Example
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
1
REVIEW •When Factoring: Write the polynomial as a product. Steps: 1. Find the GCF of all its . 2. Write the polynomial as a product by factoring out the GCF from all the . - This is done by dividing the original of the polynomial by the GCF.
3. The remaining factors in each term will form a polynomial.
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
3
The Difference of Two Squares
The Difference of Two Squares If A and B are real numbers, variables, or algebraic expressions, then
A2 B 2 A B A B . In words: The difference of the squares of two , factors as the product of a sum and a difference of those .
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
4
The Difference of Two Squares EXAMPLE
Factor: 25 x 4 9 y 6 . SOLUTION
We must express each term as the square of a monomial. Then use the formula for factoring A2 B 2 A B A B . 25 x 4 9 y 6
5x 3 y 2 2
5x
2
3 2
3 y3 5x2 3 y3
Express as the difference of two squares
Factor using the Difference of Two Squares method
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
5
The Difference of Two Squares EXAMPLE
Factor: 6 x 2 6 y 2. SOLUTION
The GCF of the two of the polynomial is 6. We begin by factoring out 6. 6x2 6 y2
6 x2 y 2
6 x y x y
Factor the GCF out of both Factor using the Difference of Two Squares method
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
6
The Difference of Two Squares EXAMPLE
Factor completely: x 4 1. SOLUTION
x 1 x 4
2 2
12
Express as the difference of two squares
x2 1 x2 1
x 2 1 x 2 12
The factors are the sum and difference of the expressions being squared
The factor x 2 1 is the difference of two squares and can be factored
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
7
The Difference of Two Squares CONTINUED
x 2 1 x 1 x 1
The factors of x 2 1 are the sum and difference of the expressions being squared
Thus, x 4 1 x 2 1 x 1 x 1 .
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
8
Factoring Completely by Grouping EXAMPLE
Factor completely: x 3 3x 2 9 x 27. SOLUTION
x 3 3x 2 9 x 27
x 3 3 x 2 9 x 27
x 2 x 3 9 x 3
x 3 x 2 9
x 3 x 3 x 3
Group with common factors Factor out the common factor from each group Factor out x + 3 from both Factor x 2 9 as the difference of two squares
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
9
Objective #1: Example “ Difference of 2 Squares
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
10
Objective #1: Example
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
11
Objective #1: You Try! Example Difference of Two Squares
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
12
Objective #1: Example
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
13
• Objective 2: Factoring Sum and Difference of 2 perfect Cubes
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
14
The Sum & Difference of Two Cubes Factoring the Sum & Difference of Two Cubes 1)
Factoring the Sum of Two Cubes
A3 B 3 A B A2 AB B 2 Same Signs
Opposite Signs
2) Factoring the Difference of Two Cubes
A3 B 3 A B A2 AB B 2 Same Signs
Opposite Signs
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
15
Write each monomial as a cube and apply either of the rules.
Example :
Rewrite as cubes 3
3
x 64 (x 4 ) 3
Apply the rule for sum of 3 3 cubes: a b ab
a ab b
2
2
2
2
(x 4)(x x 4 4 ) 2
(x 4)(x 4x 16) Copyright © 2013, 2009, 2006 Pearson Education, Inc.
16
Rewrite as cubes 3 3
3
Ex: 8y 125 ((2y) 5 ) Apply the rule for difference of cubes:
a b a ba ab b 3
3
2
2
2y 5 2y 2y 5 5 2
2
2y 54y 10y 25
2
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
17
Objective #2: Example
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
18
Objective #2 : Example
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
19
Objective #2 : Example
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
20
Objective #2 : Example
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
21
Copyright © 2013, 2009, 2006 Pearson Education, Inc.
22
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