How to Factor Trinomials

How to Factor Trinomials

 Read,3 minutes

Factoring a trinomial means rewriting a three-term polynomial as a product of simpler factors. For quadratics, the answer often looks like two binomials:

\(ax^2+bx+c=(mx+n)(px+q)\)

The big idea is that factoring is multiplication in reverse. When you multiply two binomials, the outside and inside products combine to make the middle term. Factoring asks: Which two binomials could have produced this trinomial?

Start by checking for a GCF

Before trying any trinomial pattern, look for a greatest common factor. If every term has a common factor, factor it out first. This keeps the numbers smaller and prevents missed answers.

Example: \(3x^2+12x+9=3(x^2+4x+3)=3(x+1)(x+3)\).

Case 1: Factoring \(x^2+bx+c\)

When the leading coefficient is \(1\), the binomials usually have the form \((x+m)(x+n)\). The two numbers \(m\) and \(n\) must do two jobs at the same time:

  • Multiply to the constant term: \(mn=c\)
  • Add to the middle coefficient: \(m+n=b\)

Example: factor \(x^2+7x+12\).

The factor pairs of \(12\) are \((1,12)\), \((2,6)\), and \((3,4)\). We need the pair that adds to \(7\). Since \(3+4=7\), we use \(3\) and \(4\):

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

Check: \((x+3)(x+4)=x^2+4x+3x+12=x^2+7x+12\).

How signs help you choose the numbers

  • If \(c\) is positive and \(b\) is positive, both numbers are positive.
  • If \(c\) is positive and \(b\) is negative, both numbers are negative.
  • If \(c\) is negative, one number is positive and one is negative. The larger absolute value controls the sign of \(b\).

Example: \(x^2+x-12\) needs two numbers that multiply to \(-12\) and add to \(1\). The pair \(4\) and \(-3\) works, so \(x^2+x-12=(x+4)(x-3)\).

Case 2: Factoring \(ax^2+bx+c\)

When the leading coefficient \(a\) is not \(1\), the middle term comes from both binomials, so guessing is harder. A reliable method is the AC method, also called splitting the middle term.

  1. Multiply \(a\cdot c\).
  2. Find two numbers that multiply to \(a c\) and add to \(b\).
  3. Rewrite the middle term using those two numbers.
  4. Factor by grouping.

Example: factor \(2x^2+7x+3\).

Here \(a\cdot c=2\cdot3=6\). We need two numbers that multiply to \(6\) and add to \(7\): \(6\) and \(1\).

\(2x^2+7x+3=2x^2+6x+x+3\)

Now group and factor each pair:

\((2x^2+6x)+(x+3)=2x(x+3)+1(x+3)\)

The shared binomial is \((x+3)\), so the factored form is:

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

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

Special patterns to recognize

  • Perfect square trinomial: \(a^2+2ab+b^2=(a+b)^2\) and \(a^2-2ab+b^2=(a-b)^2\).
  • Difference of squares: \(a^2-b^2=(a-b)(a+b)\). This has two terms, but it often appears in the same factoring practice.

Examples: \(4x^2-12x+9=(2x-3)^2\) and \(9x^2-25=(3x-5)(3x+5)\).

A good factoring checklist

  • First, factor out any GCF.
  • If the leading coefficient is \(1\), look for two numbers that multiply to \(c\) and add to \(b\).
  • If the leading coefficient is not \(1\), use \(a\cdot c\), split the middle term, then group.
  • Watch the signs carefully.
  • Always multiply your answer back to check it.

Free printable Worksheets

Exercises for Factoring Trinomials

1) \(x^2+7x+12\) \( \Rightarrow \)

2) \(x^2+9x+20\) \( \Rightarrow \)

3) \(x^2-5x+6\) \( \Rightarrow \)

4) \(x^2+x-12\) \( \Rightarrow \)

5) \(x^2-8x+15\) \( \Rightarrow \)

6) \(x^2-2x-24\) \( \Rightarrow \)

7) \(x^2+2x-35\) \( \Rightarrow \)

8) \(x^2-11x+30\) \( \Rightarrow \)

9) \(x^2+13x+42\) \( \Rightarrow \)

10) \(x^2-49\) \( \Rightarrow \)

11) \(2x^2+7x+3\) \( \Rightarrow \)

12) \(3x^2+10x+7\) \( \Rightarrow \)

13) \(2x^2+11x+5\) \( \Rightarrow \)

14) \(5x^2+16x+3\) \( \Rightarrow \)

15) \(6x^2+13x+6\) \( \Rightarrow \)

16) \(4x^2-12x+9\) \( \Rightarrow \)

17) \(9x^2-25\) \( \Rightarrow \)

18) \(3x^2-14x+8\) \( \Rightarrow \)

19) \(2x^2-x-15\) \( \Rightarrow \)

20) \(6x^2-x-2\) \( \Rightarrow \)

 
1) \(x^2+7x+12\) \( \Rightarrow \) \((x+3)(x+4)\). The numbers \(3\) and \(4\) multiply to \(12\) and add to \(7\).
2) \(x^2+9x+20\) \( \Rightarrow \) \((x+4)(x+5)\).
3) \(x^2-5x+6\) \( \Rightarrow \) \((x-2)(x-3)\).
4) \(x^2+x-12\) \( \Rightarrow \) \((x+4)(x-3)\).
5) \(x^2-8x+15\) \( \Rightarrow \) \((x-3)(x-5)\).
6) \(x^2-2x-24\) \( \Rightarrow \) \((x-6)(x+4)\).
7) \(x^2+2x-35\) \( \Rightarrow \) \((x+7)(x-5)\).
8) \(x^2-11x+30\) \( \Rightarrow \) \((x-5)(x-6)\).
9) \(x^2+13x+42\) \( \Rightarrow \) \((x+6)(x+7)\).
10) \(x^2-49\) \( \Rightarrow \) \((x-7)(x+7)\). Difference of squares.
11) \(2x^2+7x+3\) \( \Rightarrow \) \((2x+1)(x+3)\).
12) \(3x^2+10x+7\) \( \Rightarrow \) \((3x+7)(x+1)\).
13) \(2x^2+11x+5\) \( \Rightarrow \) \((2x+1)(x+5)\).
14) \(5x^2+16x+3\) \( \Rightarrow \) \((5x+1)(x+3)\).
15) \(6x^2+13x+6\) \( \Rightarrow \) \((3x+2)(2x+3)\).
16) \(4x^2-12x+9\) \( \Rightarrow \) \((2x-3)^2\). Perfect square trinomial.
17) \(9x^2-25\) \( \Rightarrow \) \((3x-5)(3x+5)\). Difference of squares.
18) \(3x^2-14x+8\) \( \Rightarrow \) \((3x-2)(x-4)\).
19) \(2x^2-x-15\) \( \Rightarrow \) \((2x+5)(x-3)\).
20) \(6x^2-x-2\) \( \Rightarrow \) \((3x-2)(2x+1)\).

Factoring Trinomials Practice Quiz