Factoring Trinomials

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Factoring trinomials of the form x^2 + bx + c means writing them as a product of two binomials, (x + p)(x + q). Find two numbers that multiply to c and add to b. For example, x^2 + 7x + 12 = (x + 3)(x + 4), because 3 \cdot 4 = 12 and 3 + 4 = 7. This works when the number in front of x^2 is 1.

Before this lesson: You should know multiplying binomials, factoring out the GCF, and multiplying and dividing integers.

FOIL in Reverse

Factoring a trinomial undoes FOIL. Watch what happens when you multiply (x + 3)(x + 4).

(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12

The last term, 12, is 3 times 4. The middle term, 7x, comes from 3 plus 4. So to factor, you search for the two numbers that did this.

A 2 by 2 box. Across the top: x and 4. Down the side: x and 3. The boxes are x², 4x, 3x and 12. Beside it: the two numbers 3 and 4 multiply to 12, the last term, and add to 7, the middle coefficient, so x² + 7x + 12 = (x + 3)(x + 4).
3 and 4 multiply to 12 and add to 7. They are the numbers in the two binomials.

How to Factor Trinomials

A trinomial is a polynomial with three terms. In this lesson, the leading coefficient, the number in front of x^2, is 1.

x^2 + bx + c = (x + p)(x + q) \quad \text{where } p \cdot q = c \text{ and } p + q = b

The signs of b and c tell you the signs of the two numbers.

Signs in the trinomialSigns of the two numbersExample
c positive, b positiveBoth positivex^2 + 7x + 12 = (x + 3)(x + 4)
c positive, b negativeBoth negativex^2 - 9x + 20 = (x - 4)(x - 5)
c negativeOne positive, one negative. The larger one has the sign of b.x^2 + 2x - 15 = (x + 5)(x - 3)

Steps

  1. Factor out the GCF, if there is one.
  2. Write the trinomial as x^2 + bx + c. Find b and c.
  3. List the factor pairs of c. Use the sign table to pick the signs.
  4. Find the pair that adds to b.
  5. Write the answer as (x + p)(x + q).
  6. Check by multiplying with FOIL.

Worked Examples

Example 1: Both numbers positive

Factor x^2 + 8x + 15.

You need two numbers that multiply to 15 and add to 8. Both are positive, because c and b are both positive.

Factor pair of 15Sum
1 \cdot 1516
3 \cdot 58 \;\checkmark

So x^2 + 8x + 15 = (x + 3)(x + 5).

Check: x^2 + 5x + 3x + 15 = x^2 + 8x + 15.

Example 2: Both numbers negative

Factor x^2 - 10x + 21.

The product, 21, is positive, but the sum, −10, is negative. So both numbers are negative. The pairs are -1, -21 with sum −22, and -3, -7 with sum −10.

x^2 - 10x + 21 = (x - 3)(x - 7)

Example 3: A negative last term

Factor x^2 + 3x - 28.

The product, −28, is negative, so one number is positive and one is negative. The sum is +3, so the larger number is positive.

Factor pair of −28Sum
28 \cdot (-1)27
14 \cdot (-2)12
7 \cdot (-4)3 \;\checkmark

So x^2 + 3x - 28 = (x + 7)(x - 4).

Example 4: Factor out the GCF first

Factor 2x^2 - 2x - 24.

Every term is even, so take out 2 first. Then factor the trinomial inside. You need two numbers that multiply to −12 and add to −1: they are −4 and 3.

\begin{aligned} 2x^2 - 2x - 24 &= 2(x^2 - x - 12) \\ &= 2(x - 4)(x + 3) \end{aligned}

Keep the 2 in the final answer. It is one of the factors.

Example 5: Word problem

A rectangle has an area of x^2 + 11x + 24 square inches. Its length and width are both binomials. Find them. Then find the dimensions when x = 2.

Find two numbers that multiply to 24 and add to 11. The pair is 3 and 8.

x^2 + 11x + 24 = (x + 3)(x + 8)

The width is x + 3 inches and the length is x + 8 inches. When x = 2, the rectangle is 5 inches by 10 inches. Its area is 50 square inches, and 4 + 22 + 24 = 50 checks it.

Common Mistakes

MistakeExample of the errorFix
The pair multiplies to c but doesn’t add to bx^2 + 7x + 10 = (x + 1)(x + 10)Check the sum: 2 and 5 add to 7. So (x + 2)(x + 5).
Wrong signs when b is negativex^2 - 5x + 6 = (x + 2)(x + 3)The sum must be −5, so both are negative: (x - 2)(x - 3).
Giving the larger number the wrong signx^2 - 2x - 8 = (x + 4)(x - 2)That gives +2x. Use (x - 4)(x + 2).
Not factoring completely3x^2 + 15x + 18 = (3x + 6)(x + 3)3x + 6 still has a factor of 3. Take out the GCF first: 3(x + 2)(x + 3).
Dropping the GCF2x^2 + 10x + 12 = (x + 2)(x + 3)Keep it: 2(x + 2)(x + 3).

Special Cases

Trinomials that don’t factor

Some trinomials have no pair that works. Try x^2 + 5x + 9. The pairs of 9 are 1 and 9 (sum 10) and 3 and 3 (sum 6). Neither adds to 5.

A polynomial like this is called prime. It can’t be factored using whole numbers. Writing “prime” is a correct answer.

Two variables

The same method works when the last term has a second variable. For x^2 + 5xy + 6y^2, use 2 and 3 with y attached.

x^2 + 5xy + 6y^2 = (x + 2y)(x + 3y)

When the leading coefficient is not 1

If the trinomial looks like 3x^2 + 10x + 8, and there is no GCF to take out, you need a different method. That is the next lesson, Factoring Trinomials When a Is Not 1.

Some trinomials, like x^2 + 10x + 25 = (x + 5)^2, are the special products you already know. This method still works on them.

Practice Problems

  1. Factor x^2 + 9x + 20.
  2. Factor x^2 - 8x + 15.
  3. Factor x^2 - 11x + 24.
  4. Factor x^2 + 4x - 21.
  5. Factor x^2 - 6x - 16.
  6. Factor y^2 - y - 30.
  7. Factor 3x^2 + 21x + 30.
  8. Factor x^2 + 7xy + 10y^2.
  9. Factor x^2 + 6x + 10, or explain why it can’t be factored.
  10. A rectangular poster has an area of x^2 + 14x + 45 square inches. Find binomials for its length and width. Then find its dimensions when x = 3.
Show answers
  1. (x + 4)(x + 5). 4 and 5 multiply to 20 and add to 9.
  2. (x - 3)(x - 5). Both numbers are negative.
  3. (x - 3)(x - 8)
  4. (x + 7)(x - 3). 7 and −3 multiply to −21 and add to 4.
  5. (x - 8)(x + 2). −8 and 2 multiply to −16 and add to −6.
  6. (y - 6)(y + 5). The middle coefficient is −1.
  7. 3(x^2 + 7x + 10) = 3(x + 2)(x + 5)
  8. (x + 2y)(x + 5y)
  9. It is prime. The pairs of 10 are 1 and 10 (sum 11) and 2 and 5 (sum 7). None adds to 6.
  10. (x + 5)(x + 9). When x = 3, the poster is 8 inches by 12 inches, or 96 square inches. Check: 9 + 42 + 45 = 96.

FAQ

How do you factor a trinomial when a = 1?

Find two numbers that multiply to the last term and add to the middle coefficient. Put them into (x + p)(x + q) and check with FOIL.

How do you know which signs to use?

If the last term is positive, both numbers have the sign of the middle term. If the last term is negative, one number is positive and one is negative.

What if no pair of numbers works?

Then the trinomial is prime over the integers. It can’t be factored with whole numbers, so you write “prime”.

What should you always do first?

Look for a greatest common factor. Taking it out first gives smaller numbers, like 4x^2 + 4x - 8 = 4(x^2 + x - 2).

Summary

  • Factoring x^2 + bx + c undoes FOIL.
  • Find two numbers that multiply to c and add to b.
  • If c is positive, both numbers share the sign of b. If c is negative, the signs are different.
  • Take out the GCF first, and keep it in the answer.
  • If no pair works, the trinomial is prime. Always check with FOIL.

Previous lesson: Factoring Out the Greatest Common Factor

Next lesson: Factoring Trinomials When a Is Not 1

Back to the Algebra 1 Index

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