Algebra: Greatest Common Factors
Greatest Common Factors
The greatest common factor (GCF) of a polynomial is the largest monomial that divides evenly into each term. It's very similar to the greatest common factor you calculated, except that polynomial GCFs usually contain one or more variables.
Here's how to calculate the GCF of a polynomial:
Talk the Talk
Factoring is the process of returning a polynomial product back to its original, unmultiplied pieces, called factors. The simplest technique for factoring involves identifying a polynomial's greatest common factor, the largest monomial that divides evenly into each of the polynomial's terms.
- Find the GCF of the polynomial's coefficients. This will be the coefficient of the polynomial's GCF.
- Identify common variable powers. Look at the variables in each term of the polynomial. The GCF should contain the highest possible power of every variable. Here's the catch: Every term must contain the variable raised to at least that exponent.
- Multiply. The product of steps 1 and 2 above is the GCF of the polynomial.
Once you've found the GCF of the polynomial, you can factor that polynomial. Just write the GCF followed by a set of parentheses. Inside those parentheses, you should list what's left of each polynomial term once you divide it by the GCF. In other words, the parentheses show the polynomial with the GCF "sucked out."
Example 1: Factor the polynomial 6x2y3 - 12xy2.
Solution: Start by finding the GCF of the polynomial. Its coefficient will be 6, the GCF of 6 and 12. To determine its variable part, ask yourself, "What is the maximum number of each variable that's contained in every term?" (If that doesn't work, ask yourself, "Why did I ever try to figure out algebra, anyway? It's sucking away my will to live!" and flail your arms madly in the air. It won't help you solve the problem, but you'll definitely feel better.) Look at the x's; the first term is squared, so it has two of them, but the second term only has one. Therefore, the largest number contained by both is 1, and the GCF will contain x to the power of 1.
On the other hand, both terms contain at least two y's, so the GCF will also contain a y2. Put all three pieces together to get a GCF of 6xy2. Now, divide every term by the GCF.
You don't have to use long division to get those answers. Start by dividing the coefficients. In the first term, 6 6 = 1, and in the second, -12 6 = -2. Then, apply the exponential law that states xaxb = xa - b to each term. (Subtract the denominator power from the numerator power for each matching variable.) For instance, in the first term, you'll get x2 - 1 = x1 = x and y3-2 = y1 = y.
You've Got Problems
Problem 1: Factor the polynomial 9x5y2 + 3x4y3 - 6x3y7.
You're almost done. The factored form of the original polynomial will equal the GCF times the divided form of the terms you just calculated. Just write the divided form of the terms inside parentheses and multiply that entire quantity by the GCF.
- 6xy2(xy - 2)
It's easy to check your answer. Just distribute the 6xy2 term through the parentheses, and you should end up with the original problem.
Excerpted from The Complete Idiot's Guide to Algebra 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.
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