Method

Factoring a Quadratic Trinomial

When trying to factor a quadratic trinomial of the form

a x^{2} + b x + c,

it can be difficult to see its factors. Consider the following expression.

8 x^{2} + 26 x + 6

Here,

a = 8,

b = 26,

and

c = 6 .

There are six steps to factor this trinomial.

Factor Out the GCF of

a,

b,

and

c

To fully factor a quadratic trinomial, the Greatest Common Factor (GCF) of

a,

b,

and

c

has to be factored out first. To identify the GCF of these numbers, their prime factors will be listed.

8 = 26 = 6 = 2 \cdot 2 \cdot 2 2 \cdot 13 2 \cdot 3

It can be seen that

8,

26,

and

6

share exactly one factor,

2 .

GCF (8, 26, 6) = 2

Now,

2

can be factored out.

8 x^{2} + 26 x + 6

SplitIntoFactors

Split into factors

2 (4) x^{2} + 2 (13) x + 2 (3)

FactorOut

Factor out $2$

2 (4 x^{2} + 13 x + 3)

In the remaining steps, the factored coefficient

2

before the parentheses can be ignored. The new considered quadratic trinomial is

4 x^{2} + 13 x + 3 .

Therefore, the current values of

a,

b,

and

c,

are

4,

13,

and

3,

respectively. If the GCF of the coefficients is

1,

this step can be ignored.

Find the Factor Pair of

a c

Whose Sum Is

b

It is known that $a = 4$ and $c = 3,$ so $a c = 12 > 0 .$ Therefore, the factors must have the same sign. Also, $b = 13 .$ Since the sum of the factors is positive and they must have the same sign, both factors must be positive. All positive factor pairs of $12$ can now be listed and their sums checked.

Factors of $a c$	Sum of Factors
$1$ and $12$	$1 + 12 = 13 ✓$
$2$ and $6$	$2 + 6 = 8 \times$
$3$ and $4$	$3 + 4 = 7 \times$

In this case, the correct factor pair is $1$ and $12 .$ The following table sums up how to determine the signs of the factors based on the values of $a c$ and $b .$

$a c$	$b$	Factors
Positive	Positive	Both positive
Positive	Negative	Both negative
Negative	Positive	One positive and one negative. The absolute value of the positive factor is greater.
Negative	Negative	One positive and one negative. The absolute value of the negative factor is greater.

Such analysis makes the list of possible factor pairs shorter.

Write

b x

as a Sum

The factor pair obtained in the previous step will be used to rewrite the

x -

term — the linear term — of the quadratic trinomial as a sum. Remember that the factors are

1

and

12 .

13 x

WriteSum

Write as a sum

12 x + 1 x

IdPropMult

\IdPropMult

12 x + x

The linear term

13 x

can be rewritten in the original expression as

12 x + x .

4 x^{2} + 13 x + 3 ⇕ 4 x^{2} + 12 x + x + 3

Factor Out the GCF of the First Two Terms

The expression has four terms, which can be grouped into the

first

two

and the

last

two

terms .

Then, the GCF of each group can be factored out.

4 x^{2} + 12 x + x + 3

The first two terms,

4 x^{2}

and

12 x,

can be factored.

4 x^{2} = 12 x = 2 \cdot 2 \cdot x \cdot x 2 \cdot 2 \cdot 3 \cdot x

The GCF of

4 x^{2}

and

12 x

2 \cdot 2 \cdot x = 4 x .

4 x^{2} + 12 x + x + 3

Rewrite

Rewrite $4 x^{2}$ as $4 x \cdot x$

4 x \cdot x + 12 x + x + 3

Rewrite

Rewrite $12 x$ as $4 x \cdot 3$

4 x \cdot x + 4 x \cdot 3 + x + 3

FactorOut

Factor out $4 x$

4 x (x + 3) + x + 3

Factor Out the GCF of the Last Two Terms

The process used in Step

3

will be repeated for the last two terms. In this case,

x

and

3

cannot be factored, so their GCF is

1 .

4 x (x + 3) + x + 3

Rewrite

Rewrite $x$ as $1 \cdot x$

4 x (x + 3) + 1 \cdot x + 3

Rewrite

Rewrite $3$ as $1 \cdot 3$

4 x (x + 3) + 1 \cdot x + 1 \cdot 3

FactorOut

Factor out $1$

4 x (x + 3) + 1 (x + 3)

Factor Out the Common Factor

If all the previous steps have been performed correctly, there should now be two terms with a common factor.

4 x (x + 3) + 1 (x + 3)

The common factor will be factored out.

4 x (x + 3) + 1 (x + 3)

FactorOut

Factor out $(x + 3)$

(4 x + 1) (x + 3)

The factored form of

4 x^{2} + 13 x + 3

(4 x + 1) (x + 3) .

Remember that the original trinomial was

8 x^{2} + 26 x + 6

and that the GCF

2

was factored out in Step

1 .

This GCF has to be included in the final result.

8 x^{2} + 26 x + 6 = 2 (4 x + 1) (x + 3)

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