Add and subtract Polynomials

What Are Like Terms in a Polynomial?

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

Example 1:

2 x and -6 x are like terms because they have the same variabe x to the same power 1.

Example 2:

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.

Example 3:

- x
3 and 3 x 3 are like terms because the same variable x has same power 3 in both expressions.

Example 4:

- 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.



Example 5:

- 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.

How to Group Like Terms?

Example 6: 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 7: 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 8: 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

How to Add and Subtract Polynomials?

You add polynomials by grouping like terms

Example 9: 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 10: 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 perenthesis change signs within it.

( 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 on polynomial Functions.

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