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)

and add 4 and 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}

## Add and Subtract Polynomials

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)

and add them

= 10 x

^{ 2}- 11 x + 2

### Example 6

Add , subtract and simplify:( 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 .