Date: Section: 3.1

Objectives: Recognize polynomials and their degrees.

Add, subtract, and multiply polynomials.

SM 2

Monomial: A number, a variable, or more than one number and variables multiplied together. The

variables can only have positive whole number exponents and the variables can’t be in the denominator

of a fraction or under a root.

Monomials: 2 3 415 , , , 23, ,

8 3

xyzb w x x y− Not Monomials:

1

13 54

1, , , x a z

x

−

Constant: A number by itself with no variables, like 23 or 1.−

Coefficient: The number part of a monomial (the number being multiplied by the variables.)

Polynomial: A monomial or several monomials joined by + or – signs.

Terms: The monomials that make up a polynomial. Terms are separated by + or − signs.

Like Terms: Terms whose variables and exponents are exactly the same.

Binomial: A polynomial with two unlike terms. Trinomial: A polynomial with three unlike terms.

Degree of a Monomial: The sum of the exponents on the variables.

Examples: 35x is degree 3 2 43x y is degree 6

7x− is degree 1 (It can also be written as 17x− )

25 is degree 0 (It can also be written as 025x )

Degree of a Polynomial: Find the degree of each term of the polynomial. The degree of the term with

the highest degree is the degree of the polynomial. (If there’s only one variable, it’s the highest exponent

that appears on the variable.)

Example: 6 33 6 5x x x− + The terms have degrees 6, 3, and 1, so the polynomial is degree 6

Reasons an expression isn’t a polynomial:

Examples: Decide whether each expression is a polynomial. If it is, state its degree. If it isn’t, explain

why not.

a) 4 35 2 6x x x+ + b) 54

3a a− − c)

12

2m + d) 26 1c c

−+ −

e) 1

226 5 2z z+ − f) 7 g) 8 3n− − h) 3 2x +

Adding and Subtracting Polynomials

To add or subtract polynomials, combine like terms.

• Add or subtract the coefficients. The variables and exponents do not change!

• Remember to subtract everything inside the parentheses after a minus sign. Subtract means

“add the opposite,” so change the minus sign to a plus sign and then change the signs of all the

terms inside the parentheses.

Examples: Simplify each expression.

a) ( ) ( )2 25 2 7 3n n− + − b) ( ) ( )2 22 5 4r r r r+ + −

c) ( ) ( )2 24 3 1 2 5 6x x x x− + + − + − d) ( ) ( )2 27 12 5 6 4 3z z z z+ − + − −

e) ( ) ( )2 22 3 4w w w w+ − + f) ( ) ( )3 2 2 34 2 5u u u u u− + − −

g) ( ) ( )2 26 3 2 4 3x x x x− − + − − − + h) ( ) ( )2 24 12 7 20 5 8y y y y+ − − + −

i) ( ) ( ) ( )2 2 26 5 4 2 3 7m m m m m m+ − − + − j) ( ) ( ) ( )2 22 5 3 4 8k k k k− + + − − − +

Multiplying Polynomials

To multiply two polynomials, multiply every term of one polynomial by every term of the other

polynomial. Then combine any like terms. When you are multiplying two binomials, this is sometimes

called the FOIL Method because you multiply F the first terms, O the outside terms, I the inside terms,

and L the last terms.

Examples: Multiply.

a) ( )27 3 11xy x y xy− + − b) ( ) ( )3 8m m+ −

c) ( )( )3 1 5 2x x+ − d) ( )( )2 5 5 8z z− + − −

e) ( )( )2 4 2 9t t− + f) ( )( )2 22 1 5 4u u− − +

g) ( )2

5z + h) ( )2

2 3x −

i) ( ) ( )3 3n n+ − j) ( )( )5 2 5 2y y− +

k) ( )( )22 3 5 6 7x x x− − + l) ( )( )2 24 7 3 2 8x x x x+ − − +