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Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 1
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 2
Factoring and Applications
Chapter 6
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 3
6.5
Special Factoring Techniques
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 4
6.5 Special Factoring Techniques
Objectives
1. Factor a difference of squares.
2. Factor a perfect square trinomial.
3. Factor a difference of cubes.
4. Factor a sum of cubes.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 5
Factor a difference of squares
6.5 Special Factoring Techniques
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 6
6.5 Special Factoring Techniques
Example 1 Factor each binomial, if possible.
Factor a difference of squares
(a) x2 49
(b) y2 m2 = (y – m)(y + m)
(continued)
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 7
6.5 Special Factoring Techniques
Example 1 Factor each binomial, if possible.
Factor a difference of squares
(d) x2 8
(e) p2 + 16
It is a prime polynomial because 8 is not the square of an integer.
It is a prime polynomial since it is a sum of squares.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 8
6.5 Special Factoring Techniques
Example 2 Factor each difference of squares.
Factor a difference of squares
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 9
6.5 Special Factoring Techniques
Example 3 Factor completely.
Factor a difference of squares
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 10
6.5 Special Factoring Techniques
Example 3 Factor completely.
Factor a difference of squares
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 11
Factor a perfect square trinomial.
6.5 Special Factoring Techniques
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 12
6.5 Special Factoring Techniques
Example 4 Factor x2 + 10x + 25.
Factor a perfect square trinomial.
The term x2 is a perfect square, and so is 25. Try factoring as (x + 5)2.To check, take twice the product of the two terms in the squared binomial.
Since 10x is the middle term of the trinomial, the trinomial is a perfect square and can be factored as (x + 5)2.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 13
6.5 Special Factoring Techniques
Example 5 Factor each trinomial.
Factor a perfect square trinomial.
(a) x2 – 22x + 121The first and last terms are perfect squares.Check to see if the middle term is twice the product of
the first and last terms of the binomial x – 11.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 14
6.5 Special Factoring Techniques
Example 5 Factor each trinomial.
Factor a perfect square trinomial.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 15
6.5 Special Factoring Techniques
Example 5 Factor each trinomial.
Factor a perfect square trinomial.
Twice the product of the first and last terms is 2(5y)4 = 40y, which is not the middle term.The trinomial is not a perfect square. It is a prime polynomial.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 16
6.5 Special Factoring Techniques
Example 5 Factor each trinomial.
Factor a perfect square trinomial.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 17
6.5 Special Factoring Techniques
Factor a perfect square trinomial.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 18
6.5 Special Factoring Techniques
Factor a difference of cubes.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 19
6.5 Special Factoring Techniques
Example 6a Factor the difference of cubes.
Factor a difference of cubes.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 20
6.5 Special Factoring Techniques
Example 6c Factor the difference of cubes.
Factor a difference of cubes.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 21
6.5 Special Factoring Techniques
Factor a sum of cubes.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 22
6.5 Special Factoring Techniques
Example 7a Factor the sum of cubes.
Factor a sum of cubes.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 23
6.5 Special Factoring Techniques
Example 7b Factor the sum of cubes.
Factor a sum of cubes.
Copyright © 2010 Pearson Education, Inc. All rights reserved. 6.5 – Slide 24
6.5 Special Factoring Techniques
Factor a sum of cubes.