**5.4 Factoring Trinomials**

Factoring Trinomials of the Type x^{2} + bx + c

Factoring Trinomials of the Type ax^{2} + bx + c,a ≠1

**Factoring Trinomials of the
Type x ^{2} + bx + c**

When trying to factor trinomials of the

type x^{2} + bx + c, we can use a trial-anderror

procedure.

**Constant Term Positive**

Recall the FOIL method of multiplying

two binomials:

To factor x^{2} + 7x + 10, we think of FOIL:

The first term, x^{2}, is the product of the

First terms of two binomial factors, so the

first term in each binomial must be x. The

challenge is to find two numbers p and q

such that

Thus the numbers p and q must be selected

so that their product is 10 and their sum is 7.

In this case, we know from a previous

slide that these numbers are 2 and 5. The

factorization is

**Example** Write an equivalent expression

by factoring: x^{2} + 10x + 21.

Solution

We search for factors of 21 whose sum is 10.

The factorization is thus (x + 7)(x + 3).

**Example** Write an equivalent expression

by factoring: x ^{2} – 12x + 20.

Solution

We search for factors of 20 whose sum is –12.

The factorization is thus (x – 2)(x – 10).

**Constant Term Negative**

When the constant term of a trinomial is

negative, we look for one negative factor

and one positive factor. The sum of the

factors must still be the coefficient of the

middle term.

**Example** Write an equivalent expression

by factoring: x ^{2} – x – 20.

Solution

We search for factors of –20 whose sum is –1.

The factorization is thus (x – 5)(x + 4).

Some polynomials are not factorable using

integers.

**Example** Factor: x^{2} + 2x + 4

Solution

There are no factors of 4 whose sum is 2.

This trinomial is not factorable into

binomials with integer coefficients.

**Tips for Factoring x ^{2} + bx + c**

1. If necessary, rewrite the trinomial in descending

order.

2. Find a pair of factors that have c as their product

and b as their sum. Remember the following:

. If c is positive, its factors will have the same

sign as b.

. If c is negative, one factor will be positive and

the other will be negative. Select the factors

such that the factor with the larger absolute

value is the factor with the same sign as b.

**Tips for Factoring x ^{2} + bx + c**

. If the sum of the two factors is the opposite

of b, changing the signs of both factors will

give the desired factors whose sum is b.

3. Check the result by multiplying the binomials.

**Example **Factor: x^{2} - 3xy -28y^{2}

Solution

We look for numbers p and q such that

Approach it as though it is x^{2} – 3x – 28. We look

for factors that multiply to make –28 and whose

sum is –3. Those factors are –7 and 4. Thus,

The check is left to the student.

**Factoring Trinomials of the
Type ax ^{2} + bx + c (a not 1)**

Now we look at trinomials in which

the leading coefficient is not 1.

We consider two methods.

**Method 1: Reversing FOIL**

We first consider the** FOIL method **for

factoring trinomials of the type ax^{2} + bx +c,

where a ≠1.

Consider the following multiplication.

To reverse what we did we look for two binomials

whose product is this trinomial. The product of the

First terms must be 10x^{2}. The product of the Outer

terms plus the product of the Inner terms must be 23x.

The product of the Last terms must be 12.

In general, finding such an answer involves trial

and error. We use the following method.

**To Factor ax ^{2} + bx = c by Reversing FOIL**

1. Factor out the largest common factor, if one

exists. Here we assume that none does.

2. Find two First terms whose product is ax^{2}:

3. Find two Last terms whose product is c:

4. Repeat steps (2) and (3) until a combination is

found for which the sum of the Outer and Inner

products is bx:

**Example **Factor: 2x^{2} + 7x + 6

Solution

1. Observe that there are no common factors

(other than 1 or -1).

2. To factor the first term it must be 2x times x.

(2x + )(x + )

3. The constant term 6, can be factored as

(6)(1), (3)(2), (-6)(-1), and (-3)(-2).

4. Find a pair for which the sum of the Outer

and Inner products is 7x.

Each possibility should be checked by multiplying.

This is not correct. Try again.

This is the desired factorization.

**Tips for Factoring with FOIL**

1. If the largest common factor has been factored

out of the original trinomial, then no binomial

factor can have a common factor (except 1, -1).

2. If a and c are both positive, then the signs of the

factors will be the same as the sign of b.

3. When a possible factoring produces the opposite

of the desired middle term, reverse the signs of the

constants in the factors.

4. Keep track of those possibilities that you have

tried and those you that you have not.

**Method 2: The Grouping Method**

nomials ofthe type ax^{2} + bx + c, a ≠1, is known as the

grouping method. It involves not only trial

and error and FOIL but also factoring by

grouping.

**Example** Factor: 4x^{2} + 16x + 15.

First, multiply the leading coefficient, 4, and the

constant, 15, to get 60. Then find a factorization

of 60 in which the sum of the factors is the

coefficient of the middle term: 16 (in our case 6

and 10). The middle term is then split into the

sum or difference using these factors. Then factor

by grouping.

**To Factor ax ^{2} +bx + c Using Grouping**

1. Make sure that any common factors have

been factored out.

2. Multiply the leading coefficient a and the

constant c.

3. Find a pair of factors, p and q, so that

pq = ac and p + q = b.

4. Rewrite the trinomial’s middle term, bx, as

px + qx.

5. Factor by grouping.

**Example** Factor: 6x^{2} + 5x – 6.

Solution

We look for factors of –36 that add to 5. The

factors 9 and –4 are the factors we seek.