Factoring trinomials

This worksheet is de signed to show you a few ways to factor trinomials (polynomials
with three terms). In the future, you may choose any method you desire . If you know of a
different method , tell me about it and I’ll add it .

A-C Method: This method uses “factoring by grouping”. An example of “factoring by
grouping” would be

Notice how we started with four terms. From the first two terms , we factored out 4x.
From the last two terms, we factored out -3. This gave us two terms (they are 4x(2x+1)
and 3(2x+1) ) that have a common factor . We then factored the 2x+1 out and got
(4x-3)(2x+1). Notice this results in the factored form of 8x² + 4x - 6x - 3 .

But we want to use this to factor trinomials. In general, we are factoring
ax ² + bx + c where a, b, and c are real numbers . We will find two numbers that
multiply to make ac and add to make b. Then we’ll use these two numbers to rewrite
the term bx as two terms. Then we’ll factor by grouping.

Let’s try factoring3x ² + 4x - 4 . We need two numbers that multiply to make 3(-4) or –12
and add to make 4. Think of the possible factors of –12.

There is only one pair that adds to 4. So we’ll write

Now we’ll start factoring by grouping. We’ll pull out a common factor (it’s x) from the
first two terms and another common factor (it’s 2) from the last two terms. So we have

So the factored form of 3x² + 4x - 4 is ( x + 2)(3x - 2) .

Try this guided example on your own.
Factor 3x ² -11x +10 .

Start by finding the possible factors of 3(10) or 30. (Do not forget the negatives.)

Now choose the pair that adds to –11 and rewrite the trinomial with four terms.

Now factor by grouping and finish with the factored form of 3x ² -11x +10 . Circle this
final answer. You may want to FOIL it out in your head to check your answer.

Use the A-C method to factor the following.

a.) 10x² + 4x - 6
b.) 12x² - 7x -10

Cross- product method : This is essentially a way to write the information we need in an
organized way. Let’s factor 3x ² + 4x - 4 again. We write two factors of 3x², in a column.
Then we write the factors of -4 in a second column beside the first.

We then multiply as the arrows indicate. We get 3x(1) and x(-4) or 3x and -4x. We add
these; if they add to 4x (our middle term in 3x² + 4x - 4 ) we are done. But this is not the
case (3x + -4x = -x) so we go on. Try two more factors of -4 in the second column.

Notice this time we get 3x(2) and x(-2) or 6x and –2x, which add to 4x. All we need to do
now is write the factors. Going across the top row, we get 3x + -2 or 3x – 2. Going across
the bottom row, we get x + 2. So, the factored form of 3x ² + 4x - 4 is (3x – 2)(x + 2).

Use the Cross-product method to factor the following.

a.) 6x ² + x - 12 (Hint: Remember there are essentially two ways to factor 6x². They are
6x and x or 3x and 2x. You’ll have to try them both to see which works.)
b.) 5x² + 18x - 8

Reverse FOIL
This method is also a way to write the information in an organized fashion. I call it
Reverse FOIL because it helps to understand how FOIL works when multiplying two
binomials. (A binomial is a polynomial with two terms like “x + 4”.) Consider the
multiplication problem below. Factoring goes the opposite way.

As the example illustrates, FOIL involves multiplying the First, Outside, Inside, and Last
terms of the two binomials. The First part (2x²) is the product of x and 2x. The Outside
part (-3x) is the product of x and -3. The Inside part (8x) is the product of 4 and 2x. The
Last part (-12) is the product of 4 and -3. Notice how the Inside and Outside parts add to
make the 5x in the final answer.

Now that we’ve seen how FOIL works, let’s go the opposite way. Let’s start with
2x ² + 5x -12 and see if we can factor it. We’ll start off by writing the two sets of
parentheses that we know must be a part of it.

Then we need to think about the term 2x². Again, recall this term would be formed by
multiplying the First terms in the two binomials. So let’s try 2x and x for these terms. So
we write it in.

Now, we need two numbers that multiply to make -12. These will be the Last terms in
our answer. Factors of -12 are listed below.

You can simply put each pair into the parentheses and check it to see if the pair works.
Once you find one pair that works, you can stop. So try (2x -1)(x +12) but that doesn’t
multiply to make 2x ² + 5x -12 , so that’s not right. Try (2x - 2)(x + 6) and so on. You
will find that only (2x -3)(x + 4) works.

If you want to do less trial and error, you can think about it the fol lowing way .
To make it easier to discuss, let’s write our incomplete factorization as
2x ² + 5x -12 = (2x + A)(x + B). Recall how the term 5x must be formed by the sum of
the Outside and Inside terms on the right. This means that 5x must be the sum of Ax and
2Bx. In other words “one of the factors plus twice the other should equal 5”. We see the
only pair of factors that’s true of is -3 and 4.

Try a guided example.
Factor 3x ² + 11x - 20 . First, fill in the First terms in the parentheses. They should
multiply to make 3x² and both contain an x.

Now write down the factors of -20.

Try each pair of factors in the parentheses to see which one works. Some parentheses are
provided below. Stop when you find the pair that works.

Use the Reverse FOIL method to factor these ex pressions .

a.) 2x ² - 9x -18
b.) 4x ² - 4x -15 (This is more complicated. The parentheses could be written as
(4x )(x ) or (2x )(2x ). You’ll have to try them both and see which
eventually works.)