How do you factor trinomials by grouping?

Answer 1

Factoring trinomials are no different than factorization of polynomials except for one additional step.

When factoring trinomials we're looking for the standard form of ax² + bx + c, which will later be used in determining a, b, and c for use in the quadratic formula.

Let us take a look at an example trinomial: x2+16x+32

You can see that this is a trinomial because it consists of three (tri) terms (nominal's) which are x2, 16x, and 32.

Generally, grouping requires us to have 4 terms that either have a common factor for all four terms, or a common factor that is shared between two sets of two terms. i.e. (x² and 16x both have a common factor of x, and y³, 16y², and xy² both have a common factor of y²)

Solving by trinomials by grouping requires a little more mathematical process in order to achieve our factored number. We need, for this problem, a simple mathematical equation:

ac = a+c = b

Meaning that when ever we multiply the number in the a spot and the c spot, the numbers we result in must mathematically add (or subtract) to make b.

In our earlier equation: x2+16x+32 we look at the formula ax² + bx + c.

• a = x (or 1)
• b = + 16
• c = + 32

Always pay attention to the operation signs. When dealing with trinomials there is a huge difference between x2+16x+32 and x2-16x-32.

Looking at our earlier equation: ac = a+c = b, we're saying that 1 • 32 must have some factors that when added equal too 16.

Let's take a look at 32.

• 32 • 1 = 32 but does 31 + 1 = 16? No.
• 16 • 2 = 32 but does 16 + 2 = 16? No.
• 8 • 4 = 32 but does 8 + 4 = 16? No.

Since we've tested this far and have found no numbers that can result in 16, we'd conclude that this problem is Prime. Meaning it is not able to be factored.

Now we'll take a look at a problem that can be factored.

x² - 10x + 21

Again, we need to figure out two numbers that when a • c are multiplied their addition equals to -10. Remember to keep an eye on your signs.

In our case a (1) • c (+21) = 21. Remember when I said we need to pay attention to the signs? This is very important to this problem because a • c needs to equal to -10.

21

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-1 • -21 = -21 but does -1 + -21 = -10? No.

-2, does not go into 21 evenly so we skip this number.

-3 • - 7 = 21 but does -3 + (-7) = -10? Yes.

Now that we've found, what my professor called, "the magic numbers" we need to plug them into our equation, paying attention to the signs we've used.

Our original problem was:

x² - 10x + 21

Now we can replace our "b" with -3 and -7, keeping the variables.

x² - 3x - 7x + 21

Notice anything about our problem now? Now we have the polynomial we need to be able to group and if you remembered so far that a • c = a + c = b, If we take -3x + (-7x), what do we get? -10x, our "b" number.

So we can begin factoring our problem out like any polynomial. We begin by checking to see if there is a GCF (greatest common factor) to our problem. Three of the terms have x's but the fourth does not, two of the terms have 3's as a common factor but the other two do not. So we'd say that the GCF for this problem is 1.

Break the terms down into their smallest components:

• x² = x • x
• -3x = -3 • x
• -7x = -7 • x
• +21 = 3 • 7

Looking at the first two terms x² and -3x, what terms are common to this problem?

• x² = x • x
• -3x = -3 • x

Each term has one "x" variable, this is our "outer number". x(....). We're left with, x and - 3, which is our "inside numbers" x(x - 3)

Now we look at the second two terms, -7x and +21

• -7x = -7 • x
• +21 = 3 • 7

What is common for these two? The 7. So our "outer number" is -7(....). What are we left with? 3 and x. Now, remember we want our "inner numbers" to have the same sign, which if you look at the original broken down form of:

• x • x - 3 • x - 7 • x + 3 • 7

The -7, is going to make 21 a negative when we pull it out. So we simply flip the sign inside the second parentheses from positive to negative.

Our result? 7(x - 3) - x(x - 3).

Look at these problems now as (x+a)(x+b) where (x+a) is our "inner numbers" and (x+b) is our "outer numbers". So inner numbers would be (x - 3) and our outer numbers would be (7 - x), but for mathematical appearance we're flip the outer numbers to look like our inner numbers (x - 7).

So now we're left with our factored trinomial (x - 3)(x - 7).

"But how do you know your answer is correct?" This is a very excellent question.

F.O.I.L, or First, Outer, Inner, Last terms is what we use on factors to check that our solution is correct.

First: (x - 3)(x - 7) = x • x = x²

Outer: (x - 3)(x - 7) = x • -7 = -7x

Inner: (x - 3)(x - 7) = -3 • x = -3x

Last: (x - 3)(x - 7) = -3 • -7 = +21

Put our problem back together: x² - 7x - 3x + 21, combine like terms is x² - 10x + 21.

Congratulations, you've learned how to factor trinomials and check to see if your solution is correct.