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
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) 5x2 and 7x2 are like terms because they both have an x and both x's have the same exponent
4) -2x2 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.
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 8x2. You can think of this as "3 x's + 5 x's is 8 x's"
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.
3)5x2+5x+7x2-2x= 12x2+3x This polynomial has two sets of like terms. Add them each separately. 5x2+7x2=12x2 5x-2x=3x. The final answer is 12x2+3x because 12x2 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
Answers 1) 11x-4y 2) 5y2-9y 3) 7x+4y 4) 7x 5) 5y+1 (not like terms: cannot combine)