Factor a difference of squares The formula for

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Factor a difference of squares. The formula for the product of the sum and

Factor a difference of squares. The formula for the product of the sum and difference of the same two terms is Reversing this rule leads to the following special factoring rule. For example, . The following conditions must be true for a binomial to be a difference of squares: 1. 2. Both terms of the binomial must be squares, such as x 2, 9 y 2, 25, 1, m 4. The second terms of the binomials must have different signs (one positive and one negative). Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6. 4 - 1

EXAMPLE 1 Factoring Differences of Squares Factor each binomial if possible. Solution: After any

EXAMPLE 1 Factoring Differences of Squares Factor each binomial if possible. Solution: After any common factor is removed, a sum of squares cannot be factored. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6. 4 - 2

EXAMPLE 2 Factoring Differences of Squares Factor each difference of squares. Solution: You should

EXAMPLE 2 Factoring Differences of Squares Factor each difference of squares. Solution: You should always check a factored form by multiplying. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6. 4 - 3

EXAMPLE 3 Factoring More Complex Differences of Squares Factor completely. Solution: Factor again when

EXAMPLE 3 Factoring More Complex Differences of Squares Factor completely. 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 Slide 6. 4 - 4

Factor a perfect square trinomial. The expressions 144, 4 x 2, and 81 m

Factor a perfect square trinomial. The expressions 144, 4 x 2, and 81 m 6 are called perfect squares because and 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6. 4 - 5

EXAMPLE 4 Factoring a Perfect Square Trinomial Factor k 2 + 20 k +

EXAMPLE 4 Factoring a Perfect Square Trinomial Factor k 2 + 20 k + 100. Solution: Check : Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6. 4 - 6

EXAMPLE 5 Factoring Perfect Square Trinomials Factor each trinomial. Solution: Copyright © 2008 Pearson

EXAMPLE 5 Factoring Perfect Square Trinomials Factor each trinomial. Solution: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6. 4 - 7

Factor a difference of cubes. Just as we factored the difference of squares in

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. positive same sign opposite sign This pattern for factoring a difference of cubes should be memorized. The polynomial x 3 − y 3 is not equivalent to (x − y )3, because (x − y )3 can also be written as but Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6. 4 - 8

EXAMPLE 6 Factoring Differences of Cubes Factor each polynomial. Solution: A common error in

EXAMPLE 6 Factoring Differences of Cubes Factor each polynomial. 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 + 2 xy + y 2. But because there is no 2, it is unusual to be able to further factor an expression ofas Pearson the form x 2 + xy +y 2. Slide 6. 4 - 9 Copyright © 2008 Pearson Education, Inc. Publishing Addison-Wesley

Factor a sum of cubes. A sum of squares, such as m 2 +

Factor a sum of cubes. A sum of squares, such as m 2 + 25, cannot be factored by using real numbers, but a sum of cubes can be factored by the following pattern. positive same sign opposite sign 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. ” Slide 6. 4 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The methods of factoring discussed in this section are summarized here. Copyright © 2008

The methods of factoring discussed in this section are summarized here. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6. 4 - 11

EXAMPLE 7 Factoring Sums of Cubes Factor each polynomial. Solution: Copyright © 2008 Pearson

EXAMPLE 7 Factoring Sums of Cubes Factor each polynomial. Solution: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6. 4 - 12