Section P4Polynomials

How We Describe Polynomials

n

n

The Degree of ax

If a 0, the degree of ax is n. The degree of a

nonzero constant is 0. The constant 0 has no

defined degree.

Adding and Subtracting Polynomials

Combine Like Terms

Example

Perform the indicated operations and simplify:

3 2 3 26 2 8 13 4 4 14x x x x x

Example

Perform the indicated operations and simplify:

3 2 3 29 2 9 5 8 10x x x x x x

Multiplying Polynomials

Multiplying by a monomial

Example

2 3 25 2 5 9 14x x x x

Find each product:

Multiplying Polynomials When Neither is a Monomial

Multipying each term of one polynomial by each term

of the other polynomial. Then combine like terms.

Example

24 1 10 16x x x

Find each product:

The Product of Two Binomials: FOIL

Multiplying Two Binomials

using the Distributive Property

Example

7 6 3 8x x Find each product:

Example

9 2 8 9x x

Find each product:

Multiplying the Sum and Difference of Two Terms

Example

7 4 7 4x x

Find the product:

Example

2 28 3 8 3a a

Find the product:

The Square of a Binomial

2 2

2 2 2 2

2 2 2 2

First + 2 product + Last =Product

of terms

4 x + 2 x 4 + 4 = x 8 16

2 5 2x + 2 2x -5 + -5 = 4x 20 25

x x

x x

Example

Find each product:

24x

Example

Find each product:

22 9x

Special Products

Polynomials in Several Variables

A polynomial in two variables, x and y, contains the sum of one or more monomials in the form axnym. The constant a is the coefficient. The exponents n and m represent whole numbers. The degree of a polynomial in two variables is the highest degree of all its terms.

Example

Perform the indicated operations:

3 2 3

2 2

2 4 18 19 6 7 14

5 9 7 19 4 71

x y x xy x y xy

x y xy x y xy

Example

Find the product:

3 7 8 9xy xy

Example

Find the product:

24 9x y

(a)

(b)

(c)

(d)

3 2 25 7 9 18 45x y x y xy x y xy

Perform the indicated operations.

3

3 2

3 2

3 2

4 17 52

4 19 52

5 8 17 38

5 8 19 38

x y xy

x y x y xy

x y x y xy

x y x y xy

(a)

(b)

(c)

(d)

28 9 7 8x x x

Find the product.

3 2

3 2

3 2

3

56 55 9 72

56 55 73 72

56 8 8 72

56 136

x x x

x x x

x x x

x x