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.