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# How To Factor Trinomials

Step by Step Tutorial

#### What is a Trinomial?

A trinomial is a polynomial with 3 terms.. This page will focus on quadratic trinomials. The degree of a quadratic trinomial must be '2'. In other words, there must be an exponent of '2' and that exponent must be the greatest exponent.

$$\text{Examples of Quadratic Trinomials}$$

 $$3x^2 + 2x + 1$$ $$7x^2 + 4x + 4$$ $$5 x^2 + 6x + 9$$

$$\red { \text{Non }}\text{-Examples of Quadratic Trinomials}$$

 $$x^{\red 3} + 2x + 1$$ this is not a quadratic trinomial because there is an exponent that is $$\red { \text{ greater than 2} }$$ $$2x + 4$$ this is not a quadratic trinomial because there is not exponent of 2. In fact, this is not even a trinomial because there are 2 terms $$5x ^{\red 3} + 6x^2 + 9$$ this is not a quadratic trinomial because there is an exponent that is $$\red { \text{ greater than 2} }$$
Note: For the rest of this page, 'factoring trinomials' will refer to factoring 'quadratic trinomials'. (The only difference being that a quadratic trinomial has a degree of 2.)

### Formula For Factoring Trinomials (when $$a = 1$$ )

It's always easier to understand a new concept by looking at a specific example so you might want scroll down and do that first. This formula only works when $$a = 1$$ . In other words, we will use this approach whenever the coefficient in front of x2 is 1. (If you need help factoring trinomials when $$a \ne 1$$, then go here.)

Formula Steps
1. Identify a, $$\blue b$$ , and $$\red c$$ in the trinomial $$ax^2 + \blue bx + \red c$$
2. Write down all factor pairs of $$\red c$$
3. Identify which factor pair from the previous step sum up to $$\blue b$$
4. Substitute factor pairs into two binomials

### Example of Factoring a Trinomial

Factor $$x^2 + 5x + 4$$

Step 1

Identify a, b and c in the trinomial
ax2 + bx + c

$$a = 1 \\ \blue { b = 5} \\ \red { c = 4 }$$

Step 2

Write down all factors of $$\red c$$ which multiply to $$\red { \fbox {4}}$$

(Note: since $$\red 4$$ is positive we only need to think about pairs that are either both positive or both negative. Remember a negative times a negative is a positive. As the chart on the right shows you $$-2 \cdot -2$$ is positive 4 ...so we do have to consider these negative factors.

Step 3

Identify which factor pair from the previous step sums up to $$\blue b$$

Step 4
Substitute that factor pair into two binomials

$$(x + 4)(x + 1)$$

Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

$$(x + 4)(x + 1) = x^2 + 5x + 4$$

### Practice Problems

##### Problem 1
Step 1

Identify a, b and c in the trinomial $$ax^2 + bx + c$$

$$a = 1 \\ \blue { b =4 } \\ \red{ c = 3}$$

Step 2

Write down all factor pairs of $$\red 3$$

(Note: since $$\red 3$$ is positive we only need to think about pairs that are either both positive or both negative. Remember a negative times a negative is a positive)

Step 3

Identify which factor pair from the previous step sum up to $$\blue {4 }$$

Step 4

Substitute that factor pair into two binomials

$$(x + 3)(x + 1)$$

Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

$$(x + 3)(x + 1) = x^2 + 4x + 3$$

##### Problem 2
Step 1

Identify a, b and c in the trinomial ax2 + bx + c

$$a = 1 \\ \blue { b = 5} \\ \red {c = 6}$$

Step 2

Write down all factor pairs of $$\red 6$$

Step 3

Identify which factor pair from the previous step sum up to $$\blue 5$$

Step 4

Substitute that factor pair into two binomials

(x + 2)(x + 3)

Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

(x + 2)(x + 3) = x2 + 5x + 6

##### Problem 3
Step 1

Identify a, b and c in the trinomial $$ax^2 + bx + c$$

$$a = 1 \\ \blue { b = -2} \\ \red { c= -3}$$

Step 2

Write down all factor pairs of $$\red {-3 }$$ (yes, the negative sign matters!)

(Note: since c is negative, we only need to think about pairs that have 1 negative factor and 1 positive factor. Remember a negative times a positive is a negative)

Step 3

Identify which factor pair from the previous step sums up to $$\blue { -2}$$

Step 4

Substitute that factor pair into two binomials

(x + 1)(x - 3)

Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

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

##### Problem 4
Step 1

Identify a, b and c in the trinomial $$ax^2+ bx + c$$

$$a = 1 \\ \blue{ b - 5} \\ \red{ c = 6}$$

Step 2

Write down the factor pairs of $$\red 6$$

(Note: since 6 is positive we only need to think about pairs that are either both positive or both negative. Remember a negative times a negative is a positive.)
Step 3

Identify which factor pair from the previous step sum up to $$\blue{-5}$$

Step 4

Substitute that factor pair into two binomials .

$$(x - 2)(x - 3)$$

Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

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

##### Problem 5
Step 1

Identify a, b and c in the trinomial $$ax^2 + bx + c$$

$$a = 1 \\ \blue{ b = -2 } \\ \red{ c = -15}$$

Step 2

Write down the factor pairs of $$\red{ -15}$$ (Note: since c is negative we only need to think about pairs that have 1 negative factor and 1 positive factor. Remember a negative times a positive is a negative.)

Step 3

Identify which factor pair from the previous step sums up to $$\blue{-2}$$

Step 4

Substitute that factor pair into two binomials

$$(x + 3)(x - 5)$$

Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

$$(x + 3)(x - 5) = x^2 - 2x - 15$$