# Factoring Trinomials

## What does it mean?

### Definitions:

Factoring a polynomial is expressing the polynomial as a product of two or more factors; it is somewhat the reverse process of multiplying.

To factor polynomials, we generally make use of the following properties or identities; along with other more techniques.

Perfect Square Trinomial: $$a^2 + 2ab + b^2 = ( a + b )^ 2$$ $$a^2 - 2ab + b^2 = ( a - b )^2$$

Factoring Trinomials of the form: $$x^2 + bx + c$$ A trinomial of the form is factorable over the integers, if there are two numbers p and q such that $$p * q = c \space\space and \space\space p + q = b$$ If two such numbers, p and q, exist, then the factored form of $$x^2 + bx + c = ( x + p)( x + q)$$

1. If c < 0 then p and q have different signs.
2. If c > 0 and b > 0, then p > 0 and q > 0.
3. If c > 0 and b < 0, then p < 0 and q < 0.

Factoring Trinomials of the form: $$ax^2 + bx + c$$ A trinomial of the form is factorable over the integers, if there are two numbers p and q such that $$p * q = ac \space\space and \space\space p + q = b.$$

1. Find two numbers, p and q, satisfying the two properties: $$p * q = a * c \space\space and \space\space q + p = b$$
2. Write the trinomial $$ax^2 + bx + c$$ as a four term polynomial: $$ax^2 + px + qx + c$$
3. Factor the four term polynomial using grouping: $$( ax^2 + px ) + ( qx + c )$$

## What does it look like?

Factoring Example #1$$x^2 - 2x - 8$$

First we need to find two numbers, p and q, whose product is -8 and whose sum is -2.

We know that in order for a product to be negative, then one number must be negative and one must be positive.

We get that p = 2 and q = -4.

Thus $$x^2 - 2x - 8 = ( x + 2)( x – 4 )$$ Factoring Example #2$$6x^2 - 17x + 12$$ Step 1: $$p * q = 6 * 12 \space\space and \space\space q + p = -17$$

Since b < 0 and c > 0, then p and q are both negative integers.

p q p*q = 6*12 = 72 q+p = -17
-1 -72 72 -73
-2 -36 72 -37
-3 -24 72 -27
-4 -18 72 -22
-6 -12 72 -17
-8 -9 72 -17

So, p = -8 and q = -9.

Step 2: $$6x^2 - 8x - 9x + 12$$ Step 3: $$= ( 6x^ 2 - 8x ) - ( 9x - 12 )$$ $$= 2x ( 3x - 4 ) - 3 ( 3x - 4 )$$ $$= ( 2x - 3 ) ( 3x - 4 )$$

## You'll use it...

Polynomials will show up in pretty much every section of algebra and it is important that you understand them.Trinomials will show up in pretty much every section of algebra and it is important that you understand them.

## Videos

### Example 1: Factoring trinomials with a common factor

Watch a Khan Academy Video »
Length: 5:02

### Factoring Trinomials by Grouping

Watch a Khan Academy Video »
Length: 3:46

### Factoring Trinomials With a Leading 1 Coefficient

Watch a Khan Academy Video »
Length: 4:12

### Factoring Trinomials With a Non-1 Leading Coefficient by Grouping

Watch a Khan Academy Video »
Length: 5:39