Polynomials

The Degree of axⁿ

• If a does not equal 0,the degree of axⁿ is n.
The degree of a nonzero constant is 0.The
constant 0 has no defined degree.

Definition of a Polynomial in x

• A Polynomial in x an algebraic
expression of the form

• Where and are real
numbers. ,and  n is a non-negative
integer. The Polynomial is of degree n, an
is the leading coefficient, and is the
constant term.

Text Example

Perform the indicated operations and simplify :

Solution

Group like terms .
Combine like terms .

Multiplying Polynomials

The product of two monomials is obtained by using properties of exponents .
For example

Multiply coefficients and add exponents.

Further more , we can use the distributive property to multiply a monomial and
a polynomial that is not a monomial . For example,

Multiplying Polynomials When
Neither is Monomial

• Multiply each term of one polynomial by
each term of the other polynomial. Then
combine like terms.

Using the FOIL Method to Multiply Binomials
 

Text Example

Multiply:

Solution:

Combine like terms.

The product of the Sum and
Difference of Two Terms

• The product of the sum and the difference
of the same two terms is the square of the
first term minus the square of the second
term.

The Square of a Binomial Sum

• The square of a binomial sum is the first term
squared plus 2 times the product of the
terms plus last term squared.

The Square of a Binomial
Difference

The square of a binomial difference is the first
term squared minus 2 times the product of
the terms plus last term squared.

Special Products

Let A and B represent real numbers , variables , or algebraic expression.

Special Products Sum and Difference of Two Terms	Example
Square a Binomial
Cubing a Binomial

Text Example

Multiple:

Solution

We will perform the multiplication in part (a) using the (FOIL) method. We Will
multiply in part (b) using the formula for the square of a binomial,

  Multiply these binomial using the FOIL method.

Combine like terms.

Example

Multiply:

• Solution: