Simplifying Polynomials
Polynomials must always be simplified as much as possible. That means you must add together any like terms. Like terms are terms with two things in common: 1) The same variable(s) 2) The variables have the same exponents For example: 1) 5x and 6x are like terms because they both have an x as their variables and neither has an exponent 2) 8y and 8x are not like terms because they have different variables 3) 5x^{2} and 7x^{2} are like terms because they both have an x and both x's have the same exponent 4) -2x^{2} and-5x are not like terms because they do not have the same exponent 5) 5y and 10 are not like terms because the 10 does not have a y Knowing whether or not terms are like terms is important because only like terms can be added. To add like terms: 1)Make sure the terms are like terms. If not, they cannot be added. 2)Add the coefficients (the numbers). 3)Keep the variable(s) the same. For example: 1) 3x+5x These are like terms so we can add them. Add the numbers (3+5=8) and keep the variable the same. Notice the answer is just 8x not 8x^{2}. You can think of this as "3 x's + 5 x's is 8 x's" Answer: 8x 2) 5y-7y+3=-2y+3 The 5y and -7y are like terms, so they can be added together. 5+(-7)=-2). The -2y and 3 are not like terms so you cannot add them together. Answer: -2y+3
3)5x^{2}+5x+7x^{2}-2x= 12x^{2}+3x This polynomial has two sets of like terms. Add them each separately. 5x^{2}+7x^{2}=12x^{2} 5x-2x=3x. The final answer is 12x^{2}+3x because 12x^{2} and 3x are not like terms (they have different exponents). 4) 5x+x=6x These are like terms, but in order to add them you need to know what the coefficient (number) is for the x by itself. Any variable with no coefficient has an understood coefficient of 1. This makes sense because isn't 1x the same as x just like one house is the same as a house? Now you can add them together 5x+x=6x. Practice: Simplify the following polynomials 1) 3x-4y+8x 2) -2y^{2}+7y^{2}-9y 3) 3x-5y+9y+4x 4) 3x+x+3x 5) 5y+1 Answers 1) 11x-4y 2) 5y^{2}-9y 3) 7x+4y 4) 7x 5) 5y+1 (not like terms: cannot combine) |
Related Links: Math Algebra Factors Polynomials Algebra Topics |