§

5.5

Factoring Special Forms

Special Polynomials In this section we will consider some polynomials that have special forms that make it easy for us to see how they factor. You may look at a polynomial and say, “ Oh, that ’ s just a difference of squares ” or “ I think we have a sum of cubes here.

” When you have a special polynomial, in particular one that is a difference of two squares, a perfect square polynomial, or a sum or difference of cubes, you will have a factoring formula memorized and will know how to proceed. That ’ s why these polynomials are “ special ” . They may just become our best friends among the polynomials.….

Blitzer,

Intermediate Algebra

, 5e – Slide #2 Section 5.5

The Difference of Two Squares

The Difference of Two Squares

If

A

and

B

are real numbers, variables, or algebraic expressions, then

A

2 

B

2  

A



B



A



B

 . In words: The difference of the squares of two terms, factors as the product of a sum and a difference of those terms.

Blitzer,

Intermediate Algebra

, 5e – Slide #3 Section 5.5

The Difference of Two Squares EXAMPLE Factor: 25

x

4  9

y

6

.

SOLUTION We must express each term as the square of some monomial. Then we use the formula for factoring

A

2 

B

2  

A



B



A



B

 . 25

x

4  9

y

6     2 Express as the difference of two squares  5

x

2  3

y

3  5

x

2  3

y

3  Factor using the Difference of Two Squares method Blitzer,

Intermediate Algebra

, 5e – Slide #4 Section 5.5

The Difference of Two Squares EXAMPLE Factor: 6

x

2  6

y

2

.

SOLUTION The GCF of the two terms of the polynomial is 6. We begin by factoring out 6.

6  6

x

2

x

2  

y

6 2

y

 2 Factor the GCF out of both terms 6 

x



y



x



y

 Factor using the Difference of Two Squares method Blitzer,

Intermediate Algebra

, 5e – Slide #5 Section 5.5

The Difference of Two Squares EXAMPLE Factor completely:

x

4  1

.

SOLUTION

x

4  1    2  1 2  

x

2  1 

x

2  1   

x

2  1 

x

2  1 2  Express as the difference of two squares The factors are the sum and difference of the expressions being squared 

x

2  difference of two squares and can be factored Blitzer,

Intermediate Algebra

, 5e – Slide #6 Section 5.5

The Difference of Two Squares CONTINUED  

x

2  1  

x

 1 

x

 1  

x

2  sum and difference of the expressions being squared Thus,

x

4  1  

x

2  1  

x

 1 

x

 1  . Blitzer,

Intermediate Algebra

, 5e – Slide #7 Section 5.5

Factoring Completely EXAMPLE Factor completely:

x

3  3

x

2  9

x

 27

.

SOLUTION

x

3   3

x



x

2 3   9

x

3

x

 2  27    9

x

 

x

2 

x

 3  

x

 3  27  Group terms with common factors Factor out the common factor from each group   

x

 3  

x

2 

x

 3 

x

 9   3 

x

 3  Factor out

x

+ 3 from both terms

x

2  of two squares Blitzer,

Intermediate Algebra

, 5e – Slide #8 Section 5.5

Factoring Special Forms

Factoring Perfect Square Trinomials

Let

A

and

B

be real numbers, variables, or algebraic expressions.

1 )

A

2  2

AB



B

2  

A



B

 2 2 )

A

2  2

AB



B

2  

A



B

 2 Blitzer,

Intermediate Algebra

, 5e – Slide #9 Section 5.5

Factoring Perfect Square Trinomials EXAMPLE Factor: 16

x

2  40

xy

 25

y

2

.

SOLUTION 16

x

2    2 and 25

y

2   

y

 2 be expressed as twice the product of 4

x

and -5

y

.

16

x

2  40

xy

 25

y

2    2  2    5

y

   5

y

 2   4

x

 5

y

 2 Factor Blitzer,

Intermediate Algebra

, 5e – Slide #10 Section 5.5

Grouping & Difference of Two Squares EXAMPLE Factor:

x

4 

x

2  6

x

 9

.

SOLUTION

x

4  

x

2

x

4   6

x



x

 2  9 6

x

 9  

x

4  

x

 3  2    2  

x

 3  2

x

4 Group as minus a perfect square trinomial to obtain a difference of two squares Factor the perfect square trinomial Rewrite as the difference of two squares Blitzer,

Intermediate Algebra

, 5e – Slide #11 Section 5.5

Grouping & Difference of Two Squares CONTINUED  

x

2  

x

 3   

x

2  

x

 3    

x

2 

x

 3 

x

2 

x

 3  Factor the difference of two squares. The factors are the sum and difference of the expressions being squared.

Simplify Thus,

x

4 

x

2  6

x

 9 



x

2 

x

 3



x

2 

x

 3 .



Blitzer,

Intermediate Algebra

, 5e – Slide #12 Section 5.5

The Sum & Difference of Two Cubes

Factoring the Sum & Difference of Two Cubes

1) Factoring the Sum of Two Cubes:

A

3 

B

3  

A



B





A

2 

AB



B

2



Same Signs Opposite Signs 2) Factoring the Difference of Two Cubes:

A

3 

B

3  

A



B





A

2 

AB



B

2



Same Signs Opposite Signs Blitzer,

Intermediate Algebra

, 5e – Slide #13 Section 5.5

The Sum & Difference of Two Cubes EXAMPLE Factor:

x

3

y

3  64

.

SOLUTION

x

3

y

3   64   3  4 3  

xy

  

xy

 4       4 4 



x

2

y

2   4

xy

 16



4 2  Rewrite as the Sum of Two Cubes Factor the Sum of Two Cubes Simplify Thus,

x

3

y

3  64  

xy

 4 



x

2

y

2  4

xy

 16 .



Blitzer,

Intermediate Algebra

, 5e – Slide #14 Section 5.5

The Sum & Difference of Two Cubes EXAMPLE Factor: 125

x

6  64

y

6

.

SOLUTION 125 

x

6  64

y

6

   

3 



5

x

2 



5

x

2 Rewrite as the Difference of  4

y

2 

      

2  Two Cubes Factor the Difference of Two Cubes  4

y

2



25

x

4  20

x

2

y

2  16

y

4



Simplify Thus, 125

x

6  64

y

6 



5

x

2  4

y

2



25

x

4  20

x

2

y

2  16

y

4 .



Blitzer,

Intermediate Algebra

, 5e – Slide #15 Section 5.5