Factoring Trinomials Calculator

Welcome to Omni's factoring trinomials calculator! It no only factors any quadratic trinomial but also shows you the process of factoring a trinomial step-by-step! If you want to learn how to factor trinomials by hand, scroll down and read the brief text we've prepared. There's also a bunch of examples to teach you the ac method of factoring trinomials. With Omni's help, nothing will prevent you from becoming the master of factoring quadratic trinomials! 🏆

*Do you need to draw the graph of a quadratic trinomial? Visit our dedicated parabola calculator!

Do you need to solve a quadratic equation? Try completing the square calculator!*

Recall that a quadratic trinomial is a polynomial of degree 2. We usually write quadratic trinomials in the form ax² + bx + c where a, b, c are real numbers (called coefficients) and a ≠ 0 (that is, the squared term must be present). The term a is called the leading coefficient.

If you have to factor a quadratic trinomial, then you have to determine two linear binomials such that by multiplying them you arrive at the original trinomial. That is, beginning with:

ax² + bx + c

you need to find

αx - r and βx - s,

where α, r, β, s are real numbers and α and β are non-zero, such that:

(αx - r)(βx - s) = ax² + bx + c.

As you can see, factoring quadratic trinomials is the reverse of multiplication. If you don't have much practice with multiplying binomials, we recommend you check the FOIL method calculator.

There is a special case when a quadratic trinomial arises by squaring a binomial, i.e., when the two factors of this trinomial coincide. We then say that this trinomial is a perfect square trinomial.

We have learned a lot of theory, haven't we? Is there a simpler way of factoring trinomials?, you may (and should) ask. Yes, of course there is! We created our factoring trinomials calculator to make your life easier whenever you need to factor some quadrating trinomials. Here's how to factor quadratics with the help of our factoring trinomials calculator:

1. Enter the coefficients a, b, c of the trinomial you have to factor. Do not confuse the order of the coefficients!

2. Omni's factoring trinomials calculator returns the factorization immediately and displays it beneath the coefficients you've entered.

3. Make sure to turn on the Show steps? option if you want the factoring trinomials calculator to show you the process of factoring a trinomial step-by-step.

There are several methods that you can use to factor a quadratic trinomial:

• Using the quadratic formula solver;
• Recognizing a perfect square trinomial; and
• Using the grouping method (the so-called ac method of factoring trinomials).

Let us first see an example of factoring trinomials by grouping to see how it works.

1. Let's factor the trinomial

   x² + 8x + 12.

2. We can write it as:

   x² + 2x + 6x + 12.

3. From x² + 2x we can factor out x, and from 6x + 12 we can factor out 6:

   x(x + 2) + 6(x + 2).

4. From the two summands we can factor out (x + 2):

   (x + 2)(x + 6).

That's it! We've written our trinomial as a product of two binomials.
But what exactly has happened?

As you can see, the key step was to rewrite 8x as 2x + 6x. It is far from obvious why we've chosen such a transformation. You should test similar transformations, like

8x = 3x + 5x

or

8x = 7x + x,

and see that they don't work. So, was it luck 🍀? Or magic 🧙🏻? Stay calm - it was mathematics. We will teach you how to come up with the right way of rewriting the middle term bx of a quadratic trinomial so that you can easily finish the factorization. First, however, let's summarize the procedure we've used above.

We will start with the special case of quadratic trinomials with the leading coefficient a equal to 1. That is, our trinomial is of the form x² + bx + c.

1. To correctly rewrite bx means to find two integers whose product is equal to c and whose sum is equal to b. That is, we need two integers, r and s, such that:

   r * s = c,

   r + s = b.

2. We rewrite bx as rx + sx and c as r * s, and so our trinomial takes the form:

   x² + rx + sx + r*s.

3. We factor out x from x² + rx and s from sx + r*s. We obtain:

   x(x + r) + s(x + r).

4. We factor out (x + r) from the two summands we have. This gives:

   (x + r)(x + s).

We are now ready to face the general case of factoring ax² + bx + c with any non-zero a.
First of all, be aware that sometimes we can factor out a from all three terms. The remaining trinomial will have the leading coefficient equal to 1, so we arrive at the particular case discussed above.

• For instance, if you have to factor

  3x² + 24x + 36,

  observe that

  3x² + 12x - 21 = 3(x² + 8x + 12).

  We've already found that

  x² + 8x + 12 = (x + 2)(x + 6).

  So 3x² + 24x + 36 = 3(x + 2)(x + 6).

However, it may happen that a, b, c do not have a common factor and so we have to face the challenge of factoring the trinomial
ax² + bx + c where a is not 1:

1. We need to determine two integers whose product is equal to ac and whose sum is equal to b. That is, we need two integers, r and s, such that:

   r * s = a * c,

   r + s = b.

2. We rewrite bx as rx + sx, and c as r * s / a, and so our trinomial takes the form:

   ax² + rx + sx + r*s/a.

3. We factor out ax from ax² + rx and s from sx + r*s/a. We then obtain:

   ax(x + r/a) + s(x + r/a).

4. We factor out (x + r/a) from the two summands we have. This gives:

   (x + r/a)(ax + s)

   and so we've successfully factored our trinomial.

As you can see, it is only Step 1 that can pose some problems. It's easy to tell someone: you have to find two integers such that their sum is equal to this number and their product is equal to that number. But how do you actually find these numbers? This you can learn in the next section!

Here we explain how to find two integers, r and s, such that:

r * s = a * c,

r + s = b.

In italics, we will show how to apply each step to solve the problem of factorization with a = 1, b = 8, and c = 12.

1. Compute a*c.

   We have a * c = 1 * 12 = 12.

2. List all the factors of a*c, that is, all numbers that divide a*c. You may want to use Omni's factor calculator.

   Factors of 12 are: 1, 2, 3, 4, 6, 12.

3. Based on the list of factors, write down the list of pairs of numbers whose product is a*c. Be careful to include both positive and negative numbers!

   12 = 1 * 12

   12 = 2 * 6

   12 = 3 * 4

   12 = (-1) * (-12)

   12 = (-2) * (-6)

   12 = (-3) * (-4)

4. For each element from the above list, compute the sum of factors and identify the pair that sums to b.

   1 + 12 = 13

   2 + 6 = 8

   3 + 4 = 7

   (-1) + (-12) = -13

   (-2) + (-6) = -8

   (-3) + (-4) = -7

   We see that it is the pair 2 and 6 that sums to 8.

   Actually, you may stop the computation as soon as you've found the pair that sums to b. We computed the whole list only for the sake of example.

5. The pair determined in the previous step tells you how to split bx into rx + sx.

If you want to shorten the calculations a bit, remember the following two tips:

• If a*c > 0, then r and s must have the same sign, so they are both positive or both negative. You can determine their actual sign from b:

  • If b > 0, then both r and s are positive.

  • If b > 0, then both r and s are negative.

• If a*c < 0, then r or s must have different signs, i.e., one of them is positive and the other negative.

To factor a trinomial, find its roots via the quadratic formula or use the ac method of factoring trinomials. In this way from x² + bx + c you get (x - r)(x - s), where r and s are the roots.

The ac method helps factor a quadratic trinomial into two linear terms, provided that the coefficients are integer numbers.

1. Compute the product of a and c.
2. List all the factors of a × c.
3. Find all pairs of numbers whose product is a × c. Remember the negative numbers!
4. Find the pair whose sum is equal to b.
5. Use the elements of this pair to write bx as the sum of two terms.
6. Factor out whatever you can in your trinomial.
7. You are done! Congratulations!

The ac method is also known as factoring trinomials by grouping.

No, not all trinomials can be factored. Only trinomials with two real roots can be factored. Use the discriminant to check if a trinomial can be factored.