Trinomials are expressions with 3 terms. That is where the tri- comes from. The easiest way to learn how to factor them is to put them into 1 of 2 classes.

 

Any trinomial can be written in the form

ax² + bx + c

 

For example, in the problem 2x² + 5x + 10, a = 2, b= 5 and c = 10. Recognize that x does not have to be the variable, it can be any letter. But the a has to go with the variable with an exponent of 2. The b goes with the variable with an exponent of 1 and the c is called the constant since it is not with a variable.

 

What is the most important is the leading coefficient, or the a. In the previous example it was 2. Classify all polynomials as having an a = 1 or having an a not equal to 1.

Some examples of cases where a = 1 follow.

x² + 3x + 4

1b² + 6b + 7

z² + 12z + 10

 

Some examples of cases where a is not equal to 1 follow.

5x² + 6x + 2

a = 5

 

 

3y² + y + 2

a = 3

 

The first class, a = 1, is the easiest class to work. When factoring a problem with a = 1, it factors down into (x + ?) (x + ?). Now to fill in the ?’s, look for two numbers that multiply to get c and add up to b. Look at an example to clarify this.

 

 

 

Example

 

Factor x² + 6x + 5

 

Notice that a = 1, also b = 6 and c = 5. Now try to find two numbers that multiply to get 5 (c) and add up to 6 (b). The numbers that multiply to get 5 are 1,5 and -1, -5 (multiplying 2 negatives gives a positive). When adding the first two, 1 and 5, the result is 6. When adding the second two, -1 and –5, results in - 6. Since c = 6, the first two, 1 and 5, are the ones used to put in the place of the question marks.

x² + 6x + 5 = (x + 1) (x + 5)

 

Answer!

 

It doesn’t matter which one, (x +1) or (x + 5), to put first.

 

 

 

 

Example

 

Factor x² + 4x + 4

 

Look for two numbers that multiply to c (4) and add up to b (4).

Multiples of 4 (c) are

1,4

-1, - 4

2,2

-2, -2

 

These are all the possible choices and by adding them together realize that 2, 2 are the only numbers that work (add up to b or 4).

x² + 4x + 4 = (x + 2) (x + 2)

 

Answer!

 

 

 

 

 

 

Example

 

Factor b² – 7b + 12

 

In this example, x is not the variable but b is. This does not change anything.

 

b = -7 and c = 12

 

Looking for 2 numbers that multiply to 12 and add up to –7.

 

The multiples of 12 are

12,1

3,4

-12,-1

-3,-4

6,2

-6, -2

Adding of these and realize that -3, -4 are the numbers, because they are the only two that also add up to -7

b² – 7b + 12 = (b – 3) (b – 4)

Answer!

 

Note- When factoring a trinomial that looks like x² - bx + c, the two signs in parentheses will both be negative.

 

Example

Factor x² + x – 6

b = 1 and c = -6

 

Looking for two numbers that multiply to –6 and add up to 1.

 

Here are the multiples of –6.

6,-1

3,-2

-6,1

-3,2

 

By adding all these together the only ones that add up to 1 are 3,-2.

 

x² + x – 6 = (x – 2) (x + 3)

 

Answer!

 

 

 

 

 

 

Example

 

Factor - 4 x – 21+ x²

 

Rearrange it in to the correct form to find out what a, b, and c are.

x² - 4 x – 21

 

b = - 4 and c = -21

Look for two numbers that multiply to –21 and add up to -4.

Multiples of –21 are

-21,1

-3,7

21,1

3, -7

 

3, -7 are the two numbers add up to -4 and multiply to get -21.

 

x² - 4 x – 21 = (x + 3) (x – 7)

 

Answer!

 

There is one other kind of trinomials. These are the ones in which a is not equal to 1. These usually take a lot longer and are a bit more complicated than the previous type.

The best way to do these, although it takes time, is through trial and error.

 

 

 

 

 

Example

 

7x² +11 x – 6

 

To factor, realize that a = 7, b= 11, and c = -6.

 

When it is factored correctly, it will take on this form.

 

(d x + e) (f x + g) where d,e,f, and g are numbers yet to find. Here are some rules

 

d f = a (d and f are multiples of a)

e g = c (e and g are multiples of c) and

e f + d g = b

 

With this information know that d f = 7 (because a = 7).

That means that either d is 7 and f is 1 or d is 1 and f is 7. Also the possibility exists if one substitutes –7 and –1 in for f and d.

Set d = 7 and f = 1 and see if it works.

 

Also eg = -6. Thus producing the following possibilities.

if e = -1 g = 6 (a)

or e = 6 g = -1 (b)

or e = 1 g = -6 (c)

or e = -6 g = 1 (d)

or e = 2 g = -3 (e)

or e = -3 g = 2 (f)

or e = 3 g = -2 (g)

or e = -2 g = 3 (h)

e =	g =	case
-1	6	a
6	-1	b
1	-6	c
-6	1	d
2	-3	e
-3	2	f
3	-2	g
-2	3	h

 

These are all the different possibilities. Remember that d = 7 and f = 1.

 

Look at the possibility (a) that is e = -1 and g = 6. Plug all this into the last formula (e f + d g = b).

(-1)1 + 7(6) = 41 that is not equal to b which is 11 (a)

 

Try (b).

(6)1 + 7(-1) = -1 not equal to 11 (b)

 

Keep on trying until the correct solution is found.

(1)1 + 7(-6) = -41 not equal to 11 (c)

(-6)1 + 7(-1) = -13 not equal to 11 (d)

(2)1 + 7(-3) = -19 not equal to 11 (e)

(-3)1 + 7(2) = 11 this is it!

 

b= 11

 

Stop now and know that e = -3, f = 1, d = 7, and g = 2. Plug into the formula (dx + e) (fx + g).

 

(7x – 3) (x + 2)

7x² +11 x – 6 factored is (7x – 3) (x + 2)

 

Answer!

 

 

 

 

 

 

Example

Factor -3x² +5 x – 2

 

a = -3, b = 5, and c = -2.

 

Therefore d f = -3, e g = -2 and ef + dg = 5

 

Look at the possibilities for d and f. If d = 3 f = -1, and If d = -3 f = 1. Just assign d = 3 and f = -1 and see if it works. Look at the possibilities for e and g.

 

e = 1 and g = -2 (a)

e = -1 and g = 2 (b)

e = -2 and g = 1 (c)

e = 2 and g = -1 (d)

 

Note- On the e and g combinations, look at every combination but on the d and f look at half of them. For example, plug in d = 3 and f = -1 and if the right answer is found, don’t look at d = -1 and f = 3. This is only true for d and f. Always use all combinations when looking at e and g.

 

Test for all of the possibilities. Look for when e f + d g = 5. Remember that d = 3 and f = -1.

e f + d g = 5

(1)(-1) + (3)(-2) = -7 not it (a)

(-1)(-1) + (3)(2) = 7 not it (b)

(-2)(-1) + (3)(1) = 5 Answer! (c)

 

 

Note- Keep in mind, if the correct answer is not found using d = 3 and f = -1, use d= -3 and f = 1.

 

Since e = -2, f = -1, d= 3, and g = 1, plug it in to get the answer!

 

(3 x –2) (-x + 1)

 

Answer!

 

Here is an example that combines a few of the factoring rules.

 

 

 

 

Example

 

Factor x³ + 8 x² + 7 x

 

First factor an x out of each term.

 

x(x ² + 8 x + 7)

 

Factor x ² + 8 x + 7. The multiples of 7 that add up to 8 are 7 and 1.

x (x + 7) (x + 1)

 

Answer!