Factoring Trinomials When a Is Not 1

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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.

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

  1. Factor out the GCF, if there is one.
  2. Multiply a and c.
  3. Find two numbers that multiply to ac and add to b.
  4. 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.
  5. Find the GCF of the top row. Divide to fill in the two top edges.
  6. Divide the bottom-left box by the first top edge to get the last edge.
  7. 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.

a \cdot c = 2 \cdot 3 = 6. Two numbers that multiply to 6 and add to 7 are 1 and 6. So split 7x into 6x + x.

x+3
2x2x^26x
+1x3

The top row, 2x^2 and 6x, has GCF 2x. So the top edges are x and +3. Then x \div x = 1 gives the last edge, +1.

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

Check: 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3.

Example 2: A negative middle term

Factor 5x^2 - 13x + 6.

a \cdot c = 30. The middle term is negative and 30 is positive, so both numbers are negative: -10 and -3.

x-2
5x5x^2-10x
-3-3x6
5x^2 - 13x + 6 = (5x - 3)(x - 2)

Example 3: A negative last term

Factor 6x^2 + x - 12.

a \cdot c = 6(-12) = -72. You need a product of −72 and a sum of 1. The pair is 9 and −8.

2x+3
3x6x^29x
-4-8x-12

The top row, 6x^2 and 9x, has GCF 3x. So the top edges are 2x and +3. Then -8x \div 2x = -4.

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

Example 4: Take out the GCF first

Factor 8x^2 - 20x - 12.

Every term is divisible by 4. Factor it out. Then use the AC method on 2x^2 - 5x - 3: a \cdot c = -6, and the pair is −6 and 1.

\begin{aligned} 8x^2 - 20x - 12 &= 4(2x^2 - 5x - 3) \\ &= 4(2x + 1)(x - 3) \end{aligned}

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.

a \cdot c = 72. The pair that adds to 17 is 8 and 9.

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

When x = 2, the rug is 10 feet by 7 feet, so its area is 70 square feet. Check: 6(4) + 17(2) + 12 = 70.

Common Mistakes

MistakeExample of the errorFix
Looking for a product of c instead of acFor 2x^2 + 7x + 3: no pair of 3 adds to 7Use ac = 6: 1 and 6 work.
Dropping a2x^2 + 7x + 3 = (x + 1)(x + 6)That equals x^2 + 7x + 6. The answer is (2x + 1)(x + 3).
Mixing up the positions2x^2 + 7x + 3 = (2x + 3)(x + 1)That gives 5x in the middle. FOIL to check: (2x + 1)(x + 3).
Sign errors5x^2 - 13x + 6 = (5x + 3)(x + 2)The middle term is negative, so both signs are minus: (5x - 3)(x - 2).
Not factoring completely4x^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.

TryMiddle termWorks?
(3x + 2)(x + 4)12x + 2x = 14xNo
(3x + 8)(x + 1)3x + 8x = 11xNo
(3x + 4)(x + 2)6x + 4x = 10xYes

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

  1. Factor 2x^2 + 5x + 3.
  2. Factor 3x^2 + 7x + 2.
  3. Factor 2x^2 - 9x + 10.
  4. Factor 4x^2 + 4x - 3.
  5. Factor 6x^2 - 7x - 5.
  6. Factor 5x^2 + 14x - 3.
  7. Factor 10x^2 - 11x + 3.
  8. Factor 6x^2 + 15x + 6.
  9. Factor -3x^2 + 4x + 4.
  10. 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
  1. (2x + 3)(x + 1). ac = 6, and 2 and 3 add to 5.
  2. (3x + 1)(x + 2). ac = 6, and 1 and 6 add to 7.
  3. (2x - 5)(x - 2). ac = 20, and −4 and −5 add to −9.
  4. (2x + 3)(2x - 1). ac = -12, and 6 and −2 add to 4.
  5. (3x - 5)(2x + 1). ac = -30, and −10 and 3 add to −7.
  6. (5x - 1)(x + 3). ac = -15, and 15 and −1 add to 14.
  7. (5x - 3)(2x - 1). ac = 30, and −5 and −6 add to −11.
  8. 3(2x^2 + 5x + 2) = 3(2x + 1)(x + 2)
  9. -(3x^2 - 4x - 4) = -(3x + 2)(x - 2)
  10. (4x + 3)(2x + 1). When x = 1, it is 7 meters by 3 meters, or 21 square meters. Check: 8 + 10 + 3 = 21.

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

Back to the Algebra 1 Index

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