When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, Show 44 is the GCF of 16 16 and 2020 because it is the largest number that divides evenly into both 1616 and 2020 The GCF of polynomials works the same way: 4x4x is the GCF of 16x16x and 20x220{x}^{2} because it is the largest polynomial that divides evenly into both 16x16x and 20x220{x}^{2} . A General Note: Greatest Common Factor The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials. How To: Given a polynomial expression, factor out the greatest common factor.
Example 1: Factoring the Greatest Common FactorFactor 6x3y3+45x2y2+21xy6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy . SolutionFirst, find the GCF of the expression. The GCF of 6,456,45 , and 2121 is 33 . The GCF of x3,x2{x}^{3},{x}^{2} , and xx is xx . (Note that the GCF of a set of expressions in the form xn{x}^{n} will always be the exponent of lowest degree.) And the GCF of y3,y2{y}^{3},{y}^{2} , and yy is yy . Combine these to find the GCF of the polynomial, 3xy3xy . 3xy(2x 2y2)=6x3y3,3xy(15xy)=45x2y23xy\left(2{x}^{2}{y}^{2}\right)=6{x}^{3}{y}^{3},3xy\left(15xy\right)=45{x}^{2}{y}^{2} , and 3xy(7)=21xy3xy\left(7\right)=21xy . (3xy)(2x2y2+15xy+7)\left(3xy\right)\left(2{x}^{2}{y}^{2}+15xy+7\right) Analysis of the SolutionAfter factoring, we can check our work by multiplying. Use the distributive property to confirm that (3xy)(2x2y2+15x y+7)=6x3y3+45x2y2+21xy\left(3xy\right)\left(2{x}^{2}{y}^{2}+15xy+7\right)=6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy . Try It 1Factor x(b2−a)+6(b2−a)x\left({b}^{2}-a\right)+6\left({b}^{2}-a\right) by pulling out the GCF. Licenses and AttributionsNo matter how many terms a polynomial has, you always want to check for a greatest common factor (GCF) first. If the polynomial has a GCF, factoring the rest of the polynomial is much easier because once you factor out the GCF, the remaining terms will be less cumbersome. If the GCF includes a variable, your job
becomes even easier. When solving for x in a polynomial equation, if you forget to factor out the GCF, you may miss a solution, and that could mix you up in more ways than one! Without that solution, you could end up with an incorrect graph for your polynomial. And then all your work would be for nothing! To factor the polynomial 6x4 – 12x3 + 4x2, for example, follow these steps:
About This ArticleThis article can be found in the category:
|
Factor the greatest common factor from the polynomial
Copyright © 2024 berikutyang Inc.
