 ## How to Factor Trinomials

### The Way to Factor a Trinomial in Three Simple Steps

Being able to factor a trinomial is an exceedingly vital and worthwhile skill if you are doing algebra, however, this process may additionally be quite troublesome.

### Definition of Trinomials

Trinomials are polynomials with $$3$$ terms. The first one is an $$x^2$$ term, the 2nd term is an $$x$$ term, and the 3rd term is a constant (merely a number). Trinomial: $$ax^2 \ + \ bx \ + \ c$$

### Factoring Trinomials: a = 1

With this first example, we are going to show the way to factor a trinomial whenever $$a$$, the leading coefficient is $$1$$.

### Example one:

Here we are factoring the trinomial: $$x^2 \ + \ 9x \ + \ 14$$

Step One: Determine values for $$b$$ and $$c$$. In this case, $$b \ = \ 9$$ and $$c \ = \ 14$$.

Step Two: Locate 2 two numbers which ADD to $$b$$ and MULTIPLY to $$c$$. This second step may involve a small amount of trial and error. For example, you might choose $$4$$ and $$5$$ since $$4 \ + \ 5 \ = \ 9$$. Yet $$4 \times 5$$ doesn’t come to $$28$$, therefore those numbers are no good. But, if we pick $$7$$ and $$2$$, it is simple to confirm that:
$$7 \ + \ 2 \ = \ 9$$ ($$b$$’s value); and $$7 \times 2 \ = \ 14$$ ($$c$$’s value)

Step three: Utilize the numbers you chose for writing out the factors and check. In this case, the factors are $$(x \ + \ 7)$$ and $$(x \ + \ 2)$$
Final Solution: $$(x \ + \ 7)(x \ + \ 2)$$
It is possible to check the answer via multiplying the $$2$$ factors (binomials) together to find out if the answer is the original trinomial like this:
$$x^2 \ + \ 9x \ + \ 14 \ = \ (x \ + \ 7)(x \ + \ 2) \ = \ x^2 \ + \ 7x \ + \ 2x \ + \ 14 \ = \ x^2 \ + \ 9x \ + \ 14$$
If you multiply the factors it equals the original trinomial.

### Example two:

Factor the Trinomial: $$x^2 \ + \ 3x \ - \ 18$$

Step one: Figure out values for $$b$$ and $$c$$. With this example, $$b \ = \ 3$$ and $$c \ = \ -18$$.

Step two: Locate $$2$$ numbers which ADD to $$b$$ and MULTIPLY to $$c$$. Locating the correct numbers will not always be as simple as it is with Example one.
In order to make factoring trinomials simpler, jot down every factor of $$c$$ you can think of. With this example, $$c \ = \ -18$$, therefore:
$$18 \times -1 \ = \ -18, \ 9 \times -2 \ = \ -18, \ 6 \times -3 \ = \ -18, -18 \times 1 \ = \ -18, \ ...$$
Don’t forget that the 2 numbers must multiply to $$c$$ AND add to $$b$$. The sole factors of $$-18$$ which match both those requirements are $$6$$ and $$-3$$.

Step three: Utilize the chosen numbers picked to write out the factors, then and check.
The final step is then to write out the factors: $$(x \ + \ 6)(x \ - \ 3)$$
$$x^2 \ + \ 3x \ - \ 18 \ = \ (x \ + \ 6)(x \ - \ 3)$$

### Exercises for Factoring Trinomials

1) $$2x^2 \ + \ 9x \ - \ 5 =$$

2) $$30x^2 \ + \ 14x \ - \ 8 =$$

3) $$-24x^2 \ - \ 32x \ - \ 8 =$$

4) $$8x^2 \ - \ 7x \ - \ 1 =$$

5) $$21x^2 \ + \ 5x \ - \ 4 =$$

6) $$-15x^2 \ - \ 37x \ - \ 18 =$$

7) $$15x^2 \ - \ 4x \ - \ 4 =$$

8) $$-9x^2 \ - \ 36x \ - \ 20 =$$

9) $$-9x^2 \ - \ 24x \ - \ 7 =$$

10) $$x^2 \ - \ 15 \ x \ + \ 44 \ = \$$

1) $$2x^2 \ + \ 9x \ - \ 5 =$$$$\ \color{red}{(-x \ - \ 5)(-2x \ + \ 1)}$$
2) $$30x^2 \ + \ 14x \ - \ 8 =$$$$\ \color{red}{(-10x \ - \ 8)(-3x \ + \ 1)}$$
3) $$-24x^2 \ - \ 32x \ - \ 8 =$$$$\ \color{red}{(-8x \ - \ 8)(3x \ + \ 1)}$$
4) $$8x^2 \ - \ 7x \ - \ 1 =$$$$\ \color{red}{(-8x \ - \ 1)(-x \ + \ 1)}$$
5) $$21x^2 \ + \ 5x \ - \ 4 =$$$$\ \color{red}{(-7x \ - \ 4)(-3x \ + \ 1)}$$
6) $$-15x^2 \ - \ 37x \ - \ 18 =$$$$\ \color{red}{(-5x \ - \ 9)(3x \ + \ 2)}$$
7) $$15x^2 \ - \ 4x \ - \ 4 =$$$$\ \color{red}{(-5x \ - \ 2)(-3x \ + \ 2)}$$
8) $$-9x^2 \ - \ 36x \ - \ 20 =$$$$\ \color{red}{(-3x \ - \ 10)(3x \ + \ 2)}$$
9) $$-9x^2 \ - \ 24x \ - \ 7 =$$$$\ \color{red}{(-3x \ - \ 7)(3x \ + \ 1)}$$
10) $$x^2 \ - \ 15 \ x \ + \ 44 \ = \$$$$(x \ - \ 4)(x \ - \ 11)$$

## Factoring Trinomials Practice Quiz

### TABE 11 & 12 Math Study Guide

$20.99$15.99

### SSAT Upper Level Mathematics Formulas

$6.99$5.99

### AFOQT Math Practice Workbook

$25.99$14.99

### ACT Math Practice Workbook 2022

$25.99$13.99