We add and subtract polynomials by grouping like terms. We therefore first define like terms, how to group them and then how to add and subtract them in polynomials.

## Like Terms in a Polynomial

Like terms in a polynomial are terms with the same variable(s) and the same power.

### Example 1

a) 2 x and -6 x are like terms because they have the same variable x to the same power 1.
b) 2 x
2 and 3 x are NOT like terms because the variable x has power 2 in the first expression and power 1 in the second expression.
c) - x
3 and 3 x 3 are like terms because the same variable x has same power 3 in both expressions.
d) - y x
2 and 3 x 2 are NOT like terms because the first expression has two variables x and y and the second expression has one variable x.
e) - y
2 x 4 and 3 x 4 y 2 are like terms because both expressions have same variables x and y to the same powers 4 and 2 respectively.

## Group Like Terms in a Polynomial

### Example 2

Add and simplify 4 x + 6 x
Solution
Factor x out
4 x + 6 x = x ( 4 + 6)
4 x + 6 x = 10 x

### Example 3

Add and simplify - x 3 + 9 x 3
Solution
Factor x 3 out and add coefficients
- x
3 + 9 x 3 = x 3 ( -1 + 9)
= 8 x
3

### Example 4

Add and simplify 3 y 2 x 4 - 4 x 4 y 2
Solution
Factor y 2 x 4 out and add coefficients
3 y
2 x 4 - 4 x 4 y 2 = y 2 x 4 ( 3 - 4)
= - y
2 x 4

You add polynomials by grouping like terms

### Example 5

Add and simplify the two polynomials 3 x 2 + 2 x + 7 and 7 x 2 -9 x - 5
Solution
Group like terms together
( 3 x
2 - 2 x + 7 ) + ( 7 x 2 -9 x - 5 }
= (3 x
2 + 7 x 2 ) +(- 2 x - 9 x) +(7 - 5)
= 10 x
2 - 11 x + 2

### Example 6

( 5 x
4 + 2 x 3 - 8 x 2 - 10 x + 2) + ( 7 x 3 - 9 x 2 - 5 x + 3 ) - ( - x 3 + 2 x 2 - 3 x + 7 )
Solution
Remove parentheses and if a minus sign precedes a parenthesis multiply all terms inside the parenthesis by - 1.
( 5 x
4 + 2 x 3 - 8 x 2 - 10 x + 2) + ( 7 x 3 - 9 x 2 - 5 x + 3 ) - ( - x 3 + 2 x 2 - 3 x + 7 ) = 5 x 4 + 2 x 3 - 8 x 2 - 10 x + 2 + 7 x 3 - 9 x 2 - 5 x + 3 + x 3 - 2 x 2 + 3 x - 7
Group like terms
(5 x
4 ) + (2 x 3 + 7 x 3 + x 3 ) + (- 8 x 2 - 9 x 2 - 2 x 2 ) + ( - 10 x - 5 x + 3 x ) + ( 2 + 3 - 7 )
= 5 x
4 + 10 x 3 - 19 x 2 -12 x - 2

## More References and Links to Polynomials

polynomial Functions .