Copyright © 2013, 2009, 2006 Pearson Education, Inc. Section 5.4 Factoring Trinomials Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

Factoring Trinomials In the previous section we factored certain polynomials having four terms using the method of grouping. Now, we will use trial and error and a problem solving process to factor trinomials. The polynomials we will begin with will have leading coefficients of one. We will begin by trying to factor these trinomials into a product of two binomials. Polynomials that cannot be factored over a given number set are said to be prime.

A Strategy for Factoring T Factoring Trinomials A Strategy for Factoring T 1) Enter x as the first term of each factor. 2) List pairs of factors of the constant c. 3) Try various combinations of these factors. Select the combination in which the sum of the Outside and Inside products is equal to bx. 4) Check your work by multiplying the factors using the FOIL method. You should obtain the original trinomial. If none of the possible combinations yield an Outside product and an Inside product who sum is equal to bx, the trinomial cannot be factored using integers and is called prime over the set of integers. I Sum of O + I O

Factoring Trinomials Factor: EXAMPLE Factor: SOLUTION 1) Enter x as the first term of each factor. To find the second term of each factor, we must find two integers whose product is 24 and whose sum is 11. 2) List pairs of factors of the constant, 24. Factors of 24 24, 1 –24, –1 12, 2 –12, –2 8, 3 –8, –3 6, 4 –6, –4

Factoring Trinomials CONTINUED 3) Try various combinations of these factors. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to 11x. Here is a list of the possible factorizations. Possible Factorizations of Sum of Outside and Inside Products (Should Equal 11x) (x + 24)(x + 1) x + 24x = 25x (x - 24)(x - 1) -x - 24x = -25x (x + 12)(x + 2) 2x + 12x = 14x (x - 12)(x - 2) -2x - 12x = -14x (x + 8)(x + 3) 3x + 8x = 11x (x - 8)(x - 3) -3x - 8x = -11x (x + 6)(x + 4) 4x + 6x = 10x (x - 6)(x - 4) -4x - 6x = 10x This is the required middle term.

Factoring Trinomials Thus, CONTINUED Thus, Check this result by multiplying the right side using the FOIL method. You should obtain the original trinomial. Because of the commutative property, the factorization can also be expressed as

Factoring Trinomials Factor: EXAMPLE Factor: SOLUTION The GCF of the three terms of the polynomial is 4y. Therefore, we begin by factoring out 4y. Then we factor the remaining trinomial. Factor out the GCF     Thus,

Objective #1: Example

Objective #1: Example

Objective #1: Example

Objective #1: Example

Objective #1: Example

Objective #1: Example

Objective #1: Example

Objective #1: Example

Factoring Trinomials using Substitution In some trinomials, the highest power is greater than 2, and the exponent in one of the terms is half of the other term. By letting u equal the variable to the smaller power, the trinomial can be written in a form that makes its possible factorization more obvious. If a factorization is found, we replace all occurrences of u in the factorization with the original substitution.

Objective #2: Example

Objective #2: Example

A Strategy for Factoring T Factoring Trinomials A Strategy for Factoring T Assume, for the moment, that there is no greatest common factor. 1) Find two First terms whose product is 2) Find two Last terms whose product is c: 3) By trial and error, perform steps 1 and 2 until the sum of the Outside product and Inside product is bx: I O Sum of O + I If no such combinations exist, the polynomial is prime.

Factoring Trinomials Factor: EXAMPLE Factor: SOLUTION 1) Find two First terms whose product is There is more than one choice for our two First terms. Those choices are cataloged below. ? ? 2) Find two Last terms whose product is 15. There is more than one choice for our two Last terms. Those choices are cataloged below. ? ?

Factoring Trinomials CONTINUED 3) Try various combinations of these factors. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to 19x. Here is a list of some of the possible factorizations. Possible Factorizations of Sum of Outside & Inside Products (Should Equal 19x) (6x + 1)(x + 15) 90x + x = 91x (x + 1)(6x + 15) 15x + 6x = 21x (3x + 3)(2x + 5) (2x + 3)(3x + 5) 10x + 9x = 19x (6x + 3)(x + 5) 30x + 3x = 33x (x + 3)(6x + 5) 5x + 18x = 23x This is the required middle term.

Factoring Trinomials Therefore, the factorization of is: CONTINUED Therefore, the factorization of is: (2x + 3)(3x + 5) .

Factoring Trinomials Factor: EXAMPLE Factor: SOLUTION The GCF of the three terms of the polynomial is . We begin by factoring out . 1) Find two First terms whose product is ? ?

Factoring Trinomials CONTINUED    

Factoring Trinomials CONTINUED (10y - 1)(y - 3) -30y - y = -31y Possible Factorizations of Sum of Outside & Inside Products (Should Equal -17y) (10y - 1)(y - 3) -30y - y = -31y (y - 1)(10y - 3) -3y - 10y = -13y (2y - 1)(5y - 3) -6y - 5y = -11y (5y - 1)(2y - 3) -15y - 2y = -17y This is the required middle term.  

Factoring Trinomials Factor: EXAMPLE Factor: SOLUTION Notice that the exponent on is half that of the exponent on We will let u equal the variable to the power that is half of 6. Thus, let This is the given polynomial, with written as Let Rewrite the trinomial in terms of u. Factor the trinomial. Now substitute for u. Therefore,

Objective #3: Example

Objective #3: Example

Objective #3: Example

Objective #3: Example

Objective #3: Example

Objective #3: Example

Factoring Using Grouping T Factoring Trinomials Factoring Using Grouping T 1) Multiply the leading coefficient, a, and the constant, c. 2) Find the factors of ac whose sum is b. 3) Rewrite the middle term, bx, as a sum or difference using the factors from step 2. 4) Factor by grouping.

Factoring Trinomials Factor by grouping: The trinomial is of the form EXAMPLE Factor by grouping: SOLUTION The trinomial is of the form a = 1 b = 1 c = -12  

Factoring Trinomials CONTINUED    

Factoring Trinomials 4) Factor by grouping. Group terms CONTINUED 4) Factor by grouping. Group terms Factor from each group Factor out a + 4, the common binomial factor Thus,

Objective #4: Example

Objective #4: Example

Objective #4: Example CONTINUED

Objective #4: Example CONTINUED