Chapter 6 Section 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Special Factoring Techniques Factor a difference of squares. Factor a perfect square trinomial. Factor a difference of cubes. Factor a sum of cubes

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley By reversing the rules for multiplication of binomials from Section 5.6, we get rules for factoring polynomials in certain forms. Special Factoring Techniques Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Factor a difference of squares. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factor a difference of squares. The formula for the product of the sum and difference of the same two terms is Slide Reversing this rule leads to the following special factoring rule. The following conditions must be true for a binomial to be a difference of squares: 1.Both terms of the binomial must be squares, such as x 2,9y 2,25,1,m 4. 2.The second terms of the binomials must have different signs (one positive and one negative). For example,.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Factor each binomial if possible. Solution: Factoring Differences of Squares Slide After any common factor is removed, a sum of squares cannot be factored.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factor each difference of squares. EXAMPLE 2 Factoring Differences of Squares Slide Solution: You should always check a factored form by multiplying.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Factor completely. Factoring More Complex Differences of Squares Slide Solution: Factor again when any of the factors is a difference of squares as in the last problem. Check by multiplying.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Factor a perfect square trinomial. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The expressions 144, 4x 2, and 81m 6 are called perfect squares because A perfect square trinomial is a trinomial that is the square of a binomial. A necessary condition for a trinomial to be a perfect square is that two of its terms be perfect squares. Even if two of the terms are perfect squares, the trinomial may not be a perfect square trinomial. We can multiply to see that the square of a binomial gives one of the following perfect square trinomials. Factor a perfect square trinomial. Slide and

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Factoring a Perfect Square Trinomial Slide Factor k k Check :

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Factor each trinomial. Factoring Perfect Square Trinomials Slide Solution:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1.The sign of the second term in the squared binomial is always the same as the sign of the middle term in the trinomial. Slide Factoring Perfect Square Trinomials 3.Perfect square trinomials can also be factored by using grouping or the FOIL method, although using the method of this section is often easier. 2.The first and last terms of a perfect square trinomial must be positive, because they are squares. For example, the polynomial x 2 – 2x – 1 cannot be a perfect square, because the last term is negative.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Slide Factor a difference of cubes.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factor a difference of cubes. Just as we factored the difference of squares in Objective 1, we can also factor the difference of cubes by using the following pattern. This pattern for factoring a difference of cubes should be memorized. Slide The polynomial x 3 − y 3 is not equivalent to ( x − y ) 3, because ( x − y ) 3 can also be written as but same sign positive opposite sign

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factor each polynomial. EXAMPLE 6 Factoring Differences of Cubes Slide Solution: A common error in factoring a difference of cubes, such as x 3 − y 3 = (x − y)(x 2 + xy + y 2 ), is to try to factor x 2 + xy + y 2. It is easy to confuse this factor with the perfect square trinomial x 2 + 2xy + y 2. But because there is no 2, it is unusual to be able to further factor an expression of the form x 2 + xy +y 2.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Slide Factor a sum of cubes.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Factor a sum of cubes. A sum of squares, such as m , cannot be factored by using real numbers, but a sum of cubes can be factored by the following pattern. Slide Note the similarities in the procedures for factoring a sum of cubes and a difference of cubes. 1.Both are the product of a binomial and a trinomial. 2.The binomial factor is found by remembering the “cube root, same sign, cube root.” 3.The trinomial factor is found by considering the binomial factor and remembering, “square first term, opposite of the product, square last term.” same sign positive opposite sign

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The methods of factoring discussed in this section are summarized here. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Factoring Sums of Cubes Slide Factor each polynomial. Solution: