5.4 Special Factoring Techniques

Special Factoring Techniques By reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms. Slide 5.4-3

Factor a difference of squares. Objective 1 Factor a difference of squares. Slide 5.4-4

Factor a difference of squares. The formula for the product of the sum and difference of the same two terms is Factoring a Difference of Squares For example, 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 x2, 9y2, 25, 1, m4. 2. The second terms of the binomials must have different signs (one positive and one negative). Slide 5.4-5

Factoring Differences of Squares CLASSROOM 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. Slide 5.4-6

Factoring Differences of Squares CLASSROOM EXAMPLE 2 Factoring Differences of Squares Factor each difference of squares. Solution: You should always check a factored form by multiplying. Slide 5.4-7

Factoring More Complex Differences of Squares CLASSROOM 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. Slide 5.4-8

Factor a perfect square trinomial. Objective 2 Factor a perfect square trinomial. Slide 5.4-9

Factor a perfect square trinomial. The expressions 144, 4x2, and 81m6 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. and Factoring Perfect Square Trinomials Slide 5.4-10

Factoring a Perfect Square Trinomial CLASSROOM EXAMPLE 4 Factoring a Perfect Square Trinomial Factor k2 + 20k + 100. Solution: Check : Slide 5.4-11

Factoring Perfect Square Trinomials CLASSROOM EXAMPLE 5 Factoring Perfect Square Trinomials Factor each trinomial. Solution: Slide 5.4-12

Factoring Perfect Square Trinomials 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. 2. The first and last terms of a perfect square trinomial must be positive, because they are squares. For example, the polynomial x2 – 2x – 1 cannot be a perfect square, because the last term is negative. 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. Slide 5.4-13

Factor a difference of cubes. Objective 3 Factor a difference of cubes. Slide 5.4-14

Factor a difference of cubes. Factoring a Difference of Cubes same sign positive opposite sign This pattern for factoring a difference of cubes should be memorized. The polynomial x3 − y3 is not equivalent to (x − y )3, whereas Slide 5.4-15

Factoring Differences of Cubes CLASSROOM EXAMPLE 6 Factoring Differences of Cubes Factor each polynomial. Solution: A common error in factoring a difference of cubes, such as x3 − y3 = (x − y)(x2 + xy + y2), is to try to factor x2 + xy + y2. It is easy to confuse this factor with the perfect square trinomial x2 + 2xy + y2. But because there is no 2, it is unusual to be able to further factor an expression of the form x2 + xy +y2. Slide 5.4-16

Objective 4 Factor a sum of cubes. Slide 5.4-17

Factoring a Sum of Cubes Factor a sum of cubes. A sum of squares, such as m2 + 25, cannot be factored by using real numbers, but a sum of cubes can. Factoring a Sum of Cubes same sign positive 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 5.4-18

Methods of factoring discussed in this section. Slide 5.4-19

Factoring Sums of Cubes CLASSROOM EXAMPLE 7 Factoring Sums of Cubes Factor each polynomial. Solution: Slide 5.4-20