5.3 More on Factoring Trinomials

Trinomials such as 2x2 + 7x + 6, in which the coefficient of the squared term is not 1, are factored with extensions of the methods from the previous sections.

More on Factoring Trinomials

Slide 5.3-3

Objective 1

Factor trinomials by grouping when the coefficient of the second-degree term is not 1.

Slide 5.3-4

Factor trinomials by grouping when the coefficient of the second-degree term is not 1.Recall that a trinomial such as m2 + 3m + 2 is factored by finding two numbers whose product is 2 and whose sum is 3. To factor 2x2 + 7x + 6, we look for two integers whose product is 2 · 6 = 12 and whose sum is 7.

22 7 6x x

Sum

Product is 2 · 6 = 12

Slide 5.3-5

Factor trinomials by grouping when the coefficient of the second-degree term is not 1. (cont’d)

2 22 6 2 67 3 4 .x xx x x

By considering pairs of positive integers whose product is 12, we find the necessary integers to be 3 and 4. We use these integers to write the middle term, 7x, as 7x = 3x + 4x. The trinomial 2x2 + 7x + 6 becomes

22 3 4 6x x x

2 3 2 32x x x

2 2 3x x

Slide 5.3-6

Factor.

3 1 1 1p p p

23 3 1 1p p p 23 4 1p p

212 16 3z z

2 28 6 5r rt t

3 1 1p p

Solution:

23 3 1 1p p p

212 18 2 3z z z 212 18 2 3z z z 6 2 3 1(2 3)z z z 6 1 2 3z z

2 28 10 4 5r rt rt t 2 28 10 4 5r rt rt t

2 4 5r t r t 2 4 5 1 4 5r r t t r t

Slide 5.3-7

Factoring Trinomials by GroupingCLASSROOM EXAMPLE 1

Factor 6p4 + 21p3 + 9p2.

2 23 2 6 1 3p p p p

23 2 1 3p p p

Solution:

2 23 2 7 3p p p

2 23 2 6 1 3p p p p

23 2 3 1 3p p p p

Slide 5.3-8

Factoring a Trinomial with a Common Factor by GroupingCLASSROOM EXAMPLE 2

Objective 2

Factor trinomials by using the FOIL method.

Slide 5.3-9

There is an alternative method of factoring that uses trial and error.To factor 2x2 + 7x + 6 by trial and error, we use the FOIL method in reverse, trying to find two binomials whose products work.

2 6 1x x 6x2x8x

x12x13x

2 1 6x x

If the terms of the original polynomial have greatest common factor 1, then all of that polynomials binomial factors also have GCF 1.

Incorrect Incorrect

2 3 2x x 3x4x7x Correct

Slide 5.3-10

Factor trinomials by using the FOIL method. (cont’d)

Solution:

Factor 6p2 + 19p + 10.

3 2 2 5p p

Slide 5.3-11

Factoring a Trinomial with All Positive Terms by Using FOILCLASSROOM EXAMPLE 3

Solution:

Factor 10m2 – 23m + 12.

2 3 5 4m m

Slide 5.3-12

Factoring a Trinomial with a Negative Middle Term by Using FOILCLASSROOM EXAMPLE 4

Factor 5p2 + 13p – 6.

Solution:

5 2 3p p

Slide 5.3-13

Factoring a Trinomial with a Negative Constant Term by Using FOILCLASSROOM EXAMPLE 5

Factor 6m2 + 11mn – 10n2.

3 2 2 5m n m n

Solution:

Slide 5.3-14

Factoring a Trinomial with Two VariablesCLASSROOM EXAMPLE 6

Factor.

Solution:

4 3 228 58 30x x x 3 2 224 32 6x x y xy

22 14 29 152 xx x

22 7 3 2 5x x x

2 212 16 32 x yx xy

2 6 2 3x x y x y

Slide 5.3-15

Factoring Trinomials with Common FactorsCLASSROOM EXAMPLE 7