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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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