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.

