Adding and Subtracting Polynomials
Vocabulary
A variable is a quantity represented by a letter.
A polynomial is the sum of terms that contain variables raised to
positive integer or zero powers and that have no variables in any
denominator.
A term is one of the addends in an addition expression. For example,
in the expression 2x + 4, the terms are 2x and 4.
The parts of each term that are multiplied are the factors of the term.
For example, in the term 2x from the example above, the factors are
2 and x.
Like terms have the same variable factors raised to the same powers.
For example, in the expression 2x^{2} + 3x + 7 + 3x^{2} + 4x + 9, the
2x^{2} and 3x^{2} are like terms, the 3x and 4x are like terms, and the 7
and 9 are like terms.
Adding Polynomials
To add two polynomials, use the commutative and associative
properties of addition to rewrite the sum so that like terms are
grouped, and then use the distributive property to combine like terms.
Example:
Simplify (2x + 7) + ( 4x − 9)
(2x + 7) + ( 4x − 9) 
= (2x + 7) + ( 4x + (− 9)) 

= 2x + 7 + 4x + (9) 

= 2x + 4x + 7 + (9) 

= (2 + 4)x + (2) 

= 6x  2 
Negating Polynomials
If there is a negative sign directly preceding the parenthesis
surrounding a polynomial, the negative sign applies to each term
inside the parenthesis. Use the distributive property to distribute the
negation to each term inside the parenthesis. You may think of the
negative preceding the parenthesis as a â€“1, and use the rules for
multiplying signed numbers.
Example:
Simplify.
− (5x + 7 ) = −1(5x + 7 )
− (5x + 7 ) 
= −1(5x + 7 )


= (−1)(5x) + (−1)(7)


= 5x + (7) 

= 5x + (7) 

= 5x  7 
Subtracting Polynomials
To subtract two polynomials, change the subtraction to addition of the
opposite and then add.
Example:
Simplify ( 2x + 3 )− ( 5x − 7 )
( 2x + 3 ) − ( 5x − 7 ) 
= ( 2x + 3 ) + (1)( 5x − 7 ) 

= ( 2x + 3 ) + (1)( 5x + (− 7) ) 

= ( 2x + 3 ) + (1)( 5x ) + (1)(− 7) 

= 2x + ( 5x ) + 3 + (7) 

= (3x) + (10) 

= 3x + 10 
Evaluating Polynomials
To find the value of a polynomial at a given value of the variable,
substitute the value of the variable into the polynomial everywhere
the variable appears.
