Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 1 Chapter 6 Polynomial Functions

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide Factoring Special Binomials; A Factoring Strategy

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 3 Difference of Two Squares A 2 – B 2 = (A + B)(A – B) In words, the difference of the squares of two terms is the product of the sum of the terms and the difference of the terms.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 4 Sum of Two Squares Warning In general, except for factoring out the GCF, a sum of squares cannot be factored.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 5 Example: Factoring Differences of Two Squares Factor. 1. 4x 3 – 36x 2. 16p 4 – 1

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 6 Solution 1. To factor 4x 3 – 36x, first factor out the GCF, 4x:

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 7 Solution 2. The binomial 16p 4 – 1 is a difference of squares:

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 8 Solution 2. Use a graphing calculator table to verify our work.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 9 Sum or Difference of Two Cubes

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 10 Example: Factoring a Sum or Difference of Two Cubes Factor. 1. x x 3 – 125

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 11 Solution 1. The trinomial is x 2 – 2x + 4 is prime, so we have completely factored x

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 12 Solution 2. The trinomial is x 2 + 5x + 25 is prime, so we have completely factored x 3 – 125.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 13 Example: Factoring the Difference of Two Sixth Powers Factor n 6 – p 6.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 14 Solution

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 15 Five-Step Factoring Strategy These five steps can be used to factor many polynomials (steps 2-4 can be applied to the entire polynomial or to a factor of the polynomial): 1. If the leading coefficient is positive and the GCF is not 1, factor out the GCF. If the leading coefficient in negative, factor out the opposite of the GCF.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 16 Five-Step Factoring Strategy 2. For a binomial, try using one of the properties for the difference of two squares, the sum of two cubes, or the difference of two cubes.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 17 Five-Step Factoring Strategy 3. For a trinomial of the form ax 2 + bx + c, a. If a = 1, try to find two integers whose product is c and whose sum is b. b. If a ≠ 1, try to factor by using trial and error or by grouping.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 18 Five-Step Factoring Strategy 4. For an expression with four terms, try factoring by grouping. 5. Continue applying steps 2-4 until the polynomial is completely factored.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 19 Example: Factoring a Polynomial Factor x 4 – 2x x – 2000.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 20 Solution Since the expression has four terms, try to factor it by grouping.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 21 Example: Factoring a Polynomial Factor 50t 2 w 2 – 8w 4.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 6.6, Slide 22 Solution For this expression, the GCF is 2w 2. First, factor out 2w 2 : Since the factor 25t 2 – 4w 2 has two terms, check to see whether it is the difference of two squares, which it is. So, we have