Factoring Trinomials When a Is Not 1

Factoring trinomials when a is not 1 means factoring $ax^2 + bx + c$ when the leading coefficient $a$ is not 1. Use the AC method. Multiply $a$ and $c$ , find two numbers that multiply to $ac$ and add to $b$ , then split the middle term and use a box. For example, $3x^2 + 10x + 8 = (3x + 4)(x + 2)$ .

Before this lesson: You should know factoring trinomials with $a = 1$ , factoring out the GCF, and multiplying binomials.

Extra practice: The Algebra 1 Unit 7 Practice Pack (Part 2) has printable problems, answer keys and a cheat sheet for this unit.

Why a Changes Things

When $a = 1$ , you just need two numbers that multiply to $c$ and add to $b$ . When $a$ is not 1, the first terms of the binomials also change the middle term.

Look at $(3x + 4)(x + 2)$ . The outer product is $3x \cdot 2 = 6x$ and the inner product is $4 \cdot x = 4x$ . The 3 got mixed into the middle term. That is why the simple method doesn’t work.

The AC method fixes this. The two middle pieces, $6x$ and $4x$ , multiply to $24x^2$ . That is the same as $3x^2$ times $8$ . So you look for numbers that multiply to $a \cdot c$ .

Steps beside a 2 by 2 box. a times c is 3 times 8, which is 24. The numbers 4 and 6 multiply to 24 and add to 10, so 10x splits into 6x + 4x. The box holds 3x², 6x, 4x and 8. The top row has GCF 3x, so the top labels are x and 2, and the side labels are 3x and 4. The answer is (3x + 4)(x + 2). — Split the middle term into 6x and 4x, fill the box, then find the edges. The side and top are the two factors.

The AC Method for Factoring Trinomials When a Is Not 1

Here $a$ is the leading coefficient, $b$ is the middle coefficient, and $c$ is the last term.

Steps

Factor out the GCF, if there is one.
Multiply $a$ and $c$ .
Find two numbers that multiply to $ac$ and add to $b$ .
Split the middle term into two terms using those numbers. Put $ax^2$ , the two new terms, and $c$ in a 2-by-2 box.
Find the GCF of the top row. Divide to fill in the two top edges.
Divide the bottom-left box by the first top edge to get the last edge.
Write (left edges)(top edges). Check with FOIL.

The edges work the same way as the box method in multiplying binomials, just backward.

Worked Examples

Example 1: All positive terms

Factor $2x^2 + 7x + 3$ .

Step 1. This trinomial has $a = 2$ , so the method from factoring trinomials needs one more step. Use the AC method.

2 \cdot 3 = 6

Step 2. Find $a$ times $c$ .

Step 3. Find two numbers that multiply to $6$ and add to $7$ . Try the pairs: $1$ and $6$ (sum $7$ ), $2$ and $3$ (sum $5$ ). The pair is $6$ and $1$ .

2x^2 + 7x + 3 = 2x^2 + 6x + x + 3

Step 4. Split the middle term $7x$ into $6x$ and $x$ .

Step 5. Put the four terms in a 2 by 2 box. The first term goes top left and the last term goes bottom right.

Step 6. Find the GCF of the top row, $2x^2$ and $6x$ . It is $2x$ , so write it on the left of the top row. Divide to get the top labels: $x$ and $3$ .

\frac{x}{x} = 1

Step 7. Find the bottom label. Divide the bottom-left box by the top label above it. Here is the finished box.

	$x$	$+3$
$2x$	$2x^2$	$6x$
$+1$	$x$	$3$

2x^2 + 7x + 3 = (2x + 1)(x + 3)

Step 8. The left side and the top are the two factors.

Answer: $(2x + 1)(x + 3)$

Check: Multiply the factors (see multiplying binomials): $(2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3$ .

Example 2: A negative middle term

Factor $5x^2 - 13x + 6$ .

5 \cdot 6 = 30

Step 1. Find $a$ times $c$ .

Step 2. Find two numbers that multiply to $30$ and add to $-13$ . The product is positive and the sum is negative, so both numbers are negative (see multiplying and dividing integers). Try the pairs: $-1$ and $-30$ (sum $-31$ ), $-2$ and $-15$ (sum $-17$ ), $-3$ and $-10$ (sum $-13$ ), $-5$ and $-6$ (sum $-11$ ). The pair is $-10$ and $-3$ .

5x^2 - 13x + 6 = 5x^2 - 10x - 3x + 6

Step 3. Split the middle term $-13x$ into $-10x$ and $-3x$ .

Step 4. Put the four terms in a 2 by 2 box. The first term goes top left and the last term goes bottom right.

Step 5. Find the GCF of the top row, $5x^2$ and $-10x$ . It is $5x$ , so write it on the left of the top row. Divide to get the top labels: $x$ and $-2$ .

\frac{-3x}{x} = -3

Step 6. Find the bottom label. Divide the bottom-left box by the top label above it. Here is the finished box.

	$x$	$-2$
$5x$	$5x^2$	$-10x$
$-3$	$-3x$	$6$

5x^2 - 13x + 6 = (5x - 3)(x - 2)

Step 7. The left side and the top are the two factors.

Answer: $(5x - 3)(x - 2)$

Check: $(5x - 3)(x - 2) = 5x^2 - 10x - 3x + 6 = 5x^2 - 13x + 6$ .

Example 3: A negative last term

Factor $6x^2 + x - 12$ .

6(-12) = -72

Step 1. Find $a$ times $c$ .

Step 2. Find two numbers that multiply to $-72$ and add to $1$ . The product is negative, so the signs are different. The sum is positive, so the larger number is positive. Try the pairs: $-1$ and $72$ (sum $71$ ), $-2$ and $36$ (sum $34$ ), $-3$ and $24$ (sum $21$ ), $-4$ and $18$ (sum $14$ ), $-6$ and $12$ (sum $6$ ), $-8$ and $9$ (sum $1$ ). The pair is $9$ and $-8$ .

6x^2 + x - 12 = 6x^2 + 9x - 8x - 12

Step 3. Split the middle term $x$ into $9x$ and $-8x$ .

Step 4. Put the four terms in a 2 by 2 box. The first term goes top left and the last term goes bottom right.

Step 5. Find the GCF of the top row, $6x^2$ and $9x$ . It is $3x$ , so write it on the left of the top row. Divide to get the top labels: $2x$ and $3$ .

\frac{-8x}{2x} = -4

Step 6. Find the bottom label. Divide the bottom-left box by the top label above it. Here is the finished box.

	$2x$	$+3$
$3x$	$6x^2$	$9x$
$-4$	$-8x$	$-12$

6x^2 + x - 12 = (3x - 4)(2x + 3)

Step 7. The left side and the top are the two factors.

Answer: $(3x - 4)(2x + 3)$

Check: $(3x - 4)(2x + 3) = 6x^2 + 9x - 8x - 12 = 6x^2 + x - 12$ .

Example 4: Take out the GCF first

Factor $8x^2 - 20x - 12$ .

8x^2 - 20x - 12 = 4(2x^2 - 5x - 3)

Step 1. Every term is divisible by 4. Factor it out first (see factoring out the GCF).

Step 2. Now use the AC method on $2x^2 - 5x - 3$ .

2(-3) = -6

Step 3. Find $a$ times $c$ .

Step 4. Find two numbers that multiply to $-6$ and add to $-5$ . The product is negative, so the signs are different. The sum is negative, so the larger number is negative. Try the pairs: $1$ and $-6$ (sum $-5$ ), $2$ and $-3$ (sum $-1$ ). The pair is $-6$ and $1$ .

2x^2 - 5x - 3 = 2x^2 - 6x + x - 3

Step 5. Split the middle term $-5x$ into $-6x$ and $x$ .

Step 6. Put the four terms in a 2 by 2 box. The first term goes top left and the last term goes bottom right.

Step 7. Find the GCF of the top row, $2x^2$ and $-6x$ . It is $2x$ , so write it on the left of the top row. Divide to get the top labels: $x$ and $-3$ .

\frac{x}{x} = 1

Step 8. Find the bottom label. Divide the bottom-left box by the top label above it. Here is the finished box.

	$x$	$-3$
$2x$	$2x^2$	$-6x$
$+1$	$x$	$-3$

8x^2 - 20x - 12 = 4(2x + 1)(x - 3)

Step 9. The left side and the top are the two factors. Keep the 4 in front.

Answer: $4(2x + 1)(x - 3)$

Check: $4(2x + 1)(x - 3) = 4(2x^2 - 6x + x - 3) = 4(2x^2 - 5x - 3) = 8x^2 - 20x - 12$ .

Example 5: Word problem

A rectangular rug has an area of $6x^2 + 17x + 12$ square feet. Find binomials for its length and width. Then find its size when $x = 2$ .

6 \cdot 12 = 72

Step 1. Find $a$ times $c$ .

Step 2. Find two numbers that multiply to $72$ and add to $17$ . Try the pairs: $1$ and $72$ (sum $73$ ), $2$ and $36$ (sum $38$ ), $3$ and $24$ (sum $27$ ), $4$ and $18$ (sum $22$ ), $6$ and $12$ (sum $18$ ), $8$ and $9$ (sum $17$ ). The pair is $9$ and $8$ .

6x^2 + 17x + 12 = 6x^2 + 9x + 8x + 12

Step 3. Split the middle term $17x$ into $9x$ and $8x$ .

Step 4. Put the four terms in a 2 by 2 box. The first term goes top left and the last term goes bottom right.

Step 5. Find the GCF of the top row, $6x^2$ and $9x$ . It is $3x$ , so write it on the left of the top row. Divide to get the top labels: $2x$ and $3$ .

\frac{8x}{2x} = 4

Step 6. Find the bottom label. Divide the bottom-left box by the top label above it. Here is the finished box.

	$2x$	$+3$
$3x$	$6x^2$	$9x$
$+4$	$8x$	$12$

6x^2 + 17x + 12 = (3x + 4)(2x + 3)

Step 7. The left side and the top are the two factors.

3(2) + 4 = 10

Step 8. The sides are $3x + 4$ and $2x + 3$ feet. Substitute 2 for x in the first side.

2(2) + 3 = 7

Step 9. Substitute 2 for x in the second side.

10 \cdot 7 = 70

Step 10. Multiply to find the area.

Answer: The sides are $3x + 4$ feet and $2x + 3$ feet. When $x = 2$ , the rug is 10 feet by 7 feet, or 70 square feet.

Check: Put $x = 2$ into the area: $6(2)^2 + 17(2) + 12 = 6(4) + 17(2) + 12 = 24 + 34 + 12 = 70$ .

Common Mistakes

Mistake	Example of the error	Fix
Looking for a product of $c$ instead of $ac$	For $2x^2 + 7x + 3$ : no pair of 3 adds to 7	Use $ac = 6$ : 1 and 6 work.
Dropping $a$	$2x^2 + 7x + 3 = (x + 1)(x + 6)$	That equals $x^2 + 7x + 6$ . The answer is $(2x + 1)(x + 3)$ .
Mixing up the positions	$2x^2 + 7x + 3 = (2x + 3)(x + 1)$	That gives $5x$ in the middle. FOIL to check: $(2x + 1)(x + 3)$ .
Sign errors	$5x^2 - 13x + 6 = (5x + 3)(x + 2)$	The middle term is negative, so both signs are minus: $(5x - 3)(x - 2)$ .
Not factoring completely	$4x^2 + 10x + 4 = (4x + 2)(x + 2)$	$4x + 2$ has a factor of 2: $2(2x + 1)(x + 2)$ .

Other Methods and Special Cases

Guess and check

You can also test factor pairs of $a$ and $c$ until the middle term works. For $3x^2 + 10x + 8$ , the first terms must be $3x$ and $x$ .

Try	Middle term	Works?
$(3x + 2)(x + 4)$	$12x + 2x = 14x$	No
$(3x + 8)(x + 1)$	$3x + 8x = 11x$	No
$(3x + 4)(x + 2)$	$6x + 4x = 10x$	Yes

Guess and check is fast when $a$ and $c$ have few factors. The AC method always works, so use it when you get stuck.

A negative leading coefficient

Factor out $-1$ first so that $a$ is positive: $-2x^2 + 5x + 3 = -(2x^2 - 5x - 3) = -(2x + 1)(x - 3)$ .

Prime trinomials

If no pair multiplies to $ac$ and adds to $b$ , the trinomial is prime. For $2x^2 + 3x + 4$ , $ac = 8$ . The pairs 1 and 8 (sum 9) and 2 and 4 (sum 6) don’t add to 3, so it is prime.

In the next lesson, you will see another way to finish step 4: factoring by grouping. It uses the same split.

Practice Problems

Factor $2x^2 + 5x + 3$ .
Factor $3x^2 + 7x + 2$ .
Factor $2x^2 - 9x + 10$ .
Factor $4x^2 + 4x - 3$ .
Factor $6x^2 - 7x - 5$ .
Factor $5x^2 + 14x - 3$ .
Factor $10x^2 - 11x + 3$ .
Factor $6x^2 + 15x + 6$ .
Factor $-3x^2 + 4x + 4$ .
A rectangle has an area of $8x^2 + 10x + 3$ square meters. Find binomials for its length and width. Then find its size when $x = 1$ .

Show answers

Find ac: $ac = 6$ . The pair that multiplies to $6$ and adds to $5$ is $2$ and $3$

2x^2 + 2x + 3x + 3

Split the middle term.

Box: top row $2x^2$ and $2x$ , bottom row $3x$ and $3$ . The top row’s GCF is $2x$ , so the top labels are $x$ and $1$

\frac{3x}{x} = 3

Find the bottom label: divide the bottom-left box by the top label.

(2x + 3)(x + 1)

Write the factors.

Check: $(2x + 3)(x + 1) = 2x^2 + 2x + 3x + 3 = 2x^2 + 5x + 3$

Find ac: $ac = 6$ . The pair that multiplies to $6$ and adds to $7$ is $6$ and $1$

3x^2 + 6x + x + 2

Split the middle term.

Box: top row $3x^2$ and $6x$ , bottom row $x$ and $2$ . The top row’s GCF is $3x$ , so the top labels are $x$ and $2$

\frac{x}{x} = 1

Find the bottom label: divide the bottom-left box by the top label.

(3x + 1)(x + 2)

Write the factors.

Check: $(3x + 1)(x + 2) = 3x^2 + 6x + x + 2 = 3x^2 + 7x + 2$

Find ac: $ac = 20$ . The pair that multiplies to $20$ and adds to $-9$ is $-4$ and $-5$

2x^2 - 4x - 5x + 10

Split the middle term.

Box: top row $2x^2$ and $-4x$ , bottom row $-5x$ and $10$ . The top row’s GCF is $2x$ , so the top labels are $x$ and $-2$

\frac{-5x}{x} = -5

Find the bottom label: divide the bottom-left box by the top label.

(2x - 5)(x - 2)

Write the factors.

Check: $(2x - 5)(x - 2) = 2x^2 - 4x - 5x + 10 = 2x^2 - 9x + 10$

Find ac: $ac = -12$ . The pair that multiplies to $-12$ and adds to $4$ is $-2$ and $6$

4x^2 - 2x + 6x - 3

Split the middle term.

Box: top row $4x^2$ and $-2x$ , bottom row $6x$ and $-3$ . The top row’s GCF is $2x$ , so the top labels are $2x$ and $-1$

\frac{6x}{2x} = 3

Find the bottom label: divide the bottom-left box by the top label.

(2x + 3)(2x - 1)

Write the factors.

Check: $(2x + 3)(2x - 1) = 4x^2 - 2x + 6x - 3 = 4x^2 + 4x - 3$

Find ac: $ac = -30$ . The pair that multiplies to $-30$ and adds to $-7$ is $3$ and $-10$

6x^2 + 3x - 10x - 5

Split the middle term.

Box: top row $6x^2$ and $3x$ , bottom row $-10x$ and $-5$ . The top row’s GCF is $3x$ , so the top labels are $2x$ and $1$

\frac{-10x}{2x} = -5

Find the bottom label: divide the bottom-left box by the top label.

(3x - 5)(2x + 1)

Write the factors.

Check: $(3x - 5)(2x + 1) = 6x^2 + 3x - 10x - 5 = 6x^2 - 7x - 5$

Find ac: $ac = -15$ . The pair that multiplies to $-15$ and adds to $14$ is $15$ and $-1$

5x^2 + 15x - x - 3

Split the middle term.

Box: top row $5x^2$ and $15x$ , bottom row $-x$ and $-3$ . The top row’s GCF is $5x$ , so the top labels are $x$ and $3$

\frac{-x}{x} = -1

Find the bottom label: divide the bottom-left box by the top label.

(5x - 1)(x + 3)

Write the factors.

Check: $(5x - 1)(x + 3) = 5x^2 + 15x - x - 3 = 5x^2 + 14x - 3$

Find ac: $ac = 30$ . The pair that multiplies to $30$ and adds to $-11$ is $-5$ and $-6$

10x^2 - 5x - 6x + 3

Split the middle term.

Box: top row $10x^2$ and $-5x$ , bottom row $-6x$ and $3$ . The top row’s GCF is $5x$ , so the top labels are $2x$ and $-1$

\frac{-6x}{2x} = -3

Find the bottom label: divide the bottom-left box by the top label.

(5x - 3)(2x - 1)

Write the factors.

Check: $(5x - 3)(2x - 1) = 10x^2 - 5x - 6x + 3 = 10x^2 - 11x + 3$

6x^2 + 15x + 6 = 3(2x^2 + 5x + 2)

Factor out the GCF, 3.

Find ac: $ac = 4$ . The pair that multiplies to $4$ and adds to $5$ is $4$ and $1$

2x^2 + 4x + x + 2

Split the middle term.

Box: top row $2x^2$ and $4x$ , bottom row $x$ and $2$ . The top row’s GCF is $2x$ , so the top labels are $x$ and $2$

\frac{x}{x} = 1

Find the bottom label: divide the bottom-left box by the top label.

3(2x + 1)(x + 2)

Write the factors.

Check: $3(2x + 1)(x + 2) = 3(2x^2 + 5x + 2) = 6x^2 + 15x + 6$

-3x^2 + 4x + 4 = -(3x^2 - 4x - 4)

The leading coefficient is negative, so factor out $-1$ .

Find ac: $ac = -12$ . The pair that multiplies to $-12$ and adds to $-4$ is $-6$ and $2$

3x^2 - 6x + 2x - 4

Split the middle term.

Box: top row $3x^2$ and $-6x$ , bottom row $2x$ and $-4$ . The top row’s GCF is $3x$ , so the top labels are $x$ and $-2$

\frac{2x}{x} = 2

Find the bottom label: divide the bottom-left box by the top label.

-(3x + 2)(x - 2)

Write the factors.

Check: $-(3x + 2)(x - 2) = -(3x^2 - 6x + 2x - 4) = -3x^2 + 4x + 4$

10.

Find ac: $ac = 24$ . The pair that multiplies to $24$ and adds to $10$ is $4$ and $6$

8x^2 + 4x + 6x + 3

Split the middle term.

Box: top row $8x^2$ and $4x$ , bottom row $6x$ and $3$ . The top row’s GCF is $4x$ , so the top labels are $2x$ and $1$

\frac{6x}{2x} = 3

Find the bottom label: divide the bottom-left box by the top label.

(4x + 3)(2x + 1)

Write the factors.

4(1) + 3 = 7 \text{ and } 2(1) + 1 = 3

Substitute 1 for x in each side.

7 \cdot 3 = 21

Multiply to find the area.

Check with the area expression: $8(1)^2 + 10(1) + 3 = 8 + 10 + 3 = 21$
The rectangle is 7 meters by 3 meters, or 21 square meters.

Want more practice? The Algebra 1 Unit 7 Practice Pack (Part 2) has 38 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

FAQ

How do you factor a trinomial when a is not 1?

Use the AC method. Multiply $a$ and $c$ , find two numbers that multiply to $ac$ and add to $b$ , split the middle term, and use a box or grouping to finish.

What is the AC method?

It is a way to factor $ax^2 + bx + c$ by finding two numbers with product $ac$ and sum $b$ . Those numbers tell you how to split the middle term.

Can I just use guess and check?

Yes. Test factor pairs of the first and last terms until FOIL gives the right middle term. It works well for small numbers.

What if a is negative?

Factor out $-1$ first. Then factor the trinomial inside, and keep the negative sign in front of your answer.

Summary

When $a$ is not 1, look for two numbers that multiply to $ac$ and add to $b$ .
Use those numbers to split the middle term into two terms.
Put the four terms in a box. The edges of the box are the factors.
Take out the GCF first, and factor out −1 if the leading coefficient is negative.
Always check with FOIL. If no pair works, the trinomial is prime.

Previous lesson: Factoring Trinomials

Next lesson: Factoring by Grouping

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