Factoring Out the Greatest Common Factor (GCF)
Now that we know how to find the greatest common factor of monomials, we can take the first step in factoring a polynomial.
If I give you a polynomial and ask you to factor it, the first thing you need to do is look at all of the terms and find their greatest common factor. (If there is one.) Once you've found it, you can write every term as a product of the GCF and another monomial and then factor the GCF out. Here's how it works in practice:
Factor: | 3x^{3}y + 6x^{2}y | |
1. Find the greatest common factor of the polynomial's terms. The polynomial's terms are 3x^{3}y and 6x^{2}y. Applying the techniques that we learned in the "Common Factors" lesson, we can see that the greatest common factor of those monomials is 3x^{2}y. | 3xy^{2} | |
2. Rewrite each of the terms using the greatest common factor. Take a close look at how this step works - it's the key one in the process. Notice how 3x^{2}y · x = 3x^{2}y · x = 3x^{3}y which is the first term in the original polynomial and 3x^{2}y · 2 = 6x^{2}y which is the original second term. | 3x^{2}y · x + 3x^{2}y · 2 | |
3. Factor the GCF out of every term. This is going to give us our final answer. | 3x^{2}y(x + 2) |
Example 1
Simplify 6x^{3} + 2x^{2} - 2x.
The three terms we have to look at are 6x^{3}, 2x^{2} and 2x. (For this part of the discussion we don't need to worry about the negative sign.) Applying the techniques of the "Common Factors" lession, the greatest common factor of those three terms is 2x.
This brings us to step two. We need to rewrite each of the three terms as something times the greatest common facor. That would look like:
2x · 3x^{2} + 2x · x - 2x · 1
Notice how I put a 1 on the last term. A term is never going to go away because of factoring out the greatest common factor. There has to be something left behind, even if it's just a 1.
Now I'm going to factor out the 2x and leave everything else behind, including the negative sign.
2x (3x^{2} + x - 1)
Example 3
Simplify 8m^{2}n^{2}o - 4mn^{2}o + 20m^{2}n^{3}o^{3}.
This problem follows the same pattern we've seen in previous examples. First we find the greatest common factor of 8m^{2}n^{2}o, 4mn^{2}o and 20m^{2}n^{3}o^{3}. Using the techniques of "Common Factors" we can see that the greatest common factor is 4mn^{2}o. If I rewrite all the terms of the original polynomial in terms of 4mn^{2}o, I get:
4mn^{2}o · 2m - 4mn^{2}o · 1 + 4mn^{2}o · 5mno^{2}
Now I'll factor out the 4mn^{2}o and be left with the factored version.
4mn^{2}o(2m - 1 + 5mno^{2})
Example 2
Simplify 12xy^{3} - 16x^{4}y^{3}.
Applying the techniques of the "Common Factors" lesson, we can see that the greatest common factor of 12xy^{3} and 16x^{4}y^{3} is 4xy^{3}. If I rewrite each of the terms in terms of 4xy^{3} it looks like:
4xy^{3} · 3 - 4xy^{3} · 4x^{3}
Now if we factor out the 4xy^{3} we get the final factorization.
4xy^{3} (3 - 4x^{3})
Example 4
Factor -81ab^{2} - 12ab^{2}c - 3bc^{3}.
This one starts out much like the others but it has a little twist at the end. First, we need the greatest common factor of 81ab^{2}, 12ab^{2}c and 3bc^{3}. Applying the techniques of the "Common Factors" lesson, we can see that that's 3b. Rewriting each of the terms in the original expression in terms of 3b gives us:
-3b · 27ab - 3b · 4abc - 3b · c^{3}
Notice how, once again, I left all the negative signs where they were. Now I'm going to factor out all of the 3b's to get
3b(-27ab - 4abc - c^{3})
That's a perfectly good factorization and there'd be nothing wrong with stopping there. However, if you'll look closely, you'll see that all three of the terms have a negative sign. Whenever all three of the terms have something, you should always ask yourself if it's possible to factor it out. In this case we could simplify the expression a little further by pulling out the negative sign. We'll follow the same pattern here that we did with the 3b. First, I'll write every term in terms of -1. Pay careful attention to how I handle the plus signs.
3b(-1 · 27ab + (-1) · 4abc + (-1) · c^{3})
Now I can factor the negative one out to get:
(-1) · 3b(27ab + 4abc + c^{3})
-3b(27ab + 4abc + c^{3})
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Directions: This solution has 6 steps. To see a description of each step click on the boxes on the left side below. To see the calculations, click on the corresponding box on the right side. Try working out the solution yourself and use the descriptions if you need a hint and the calculations to check your solution.