# Polynomials

Recall that a monomial is a number, a variable, or a product of numbers andvariables. A polynomial is a monomial or a sum of monomials. The exponents of the variables of a polynomial must be positive. A binomial isthe sum of two monomials, and a trinomial is the sum of three monomials. The degree of a monomial is the sum of the exponents of its variables. To find the degree of a polynomial, you must find the degree of each term. The greatest degree of any term is the degree of the polynomial. The terms of a polynomial are usually arranged so that the powers of one variable are in ascending or descending order.

Examples

Consider the expression .

A Is the expression a polynomial and if so is it a monomial, binomial, or trinomial?

The expression is the sum of three monomials, therefore it is a polynomial. Since there are three monomials, the polynomial is a trinomial.

B What is the degree of the polynomial?

The degree of is 2, the degree of 5 is 0, and the degree of 7x is 1. The greatest degree is 2, so the degree of the polynomial is 2.

C Arrange the terms of the polynomial so thatthe powers of x are in descending order.

To add polynomials, you can group like terms and then find their sum, or youcan write them in column form and then add. To subtract a polynomial, add its additive inverse, which is the opposite of each term in the polynomial.

Examples

Find each sum or difference.

A

B (12x + 7y ) - (- x + 2y )

Find the additive inverse of - x + 2y. Then group the like terms and add. The additive inverse of - x + 2y is x - 2y.

(12x + 7y ) - (- x + 2y )

= (12x + 7y ) + (+ x - 2y )

= (12x + x) + (7y - 2y)

= 13x + 5y

## Multiplying a Polynomial by a Monomial

Use the distributive property to multiply a polynomial by a monomial. Youmay find it easier to multiply a polynomial by a monomial if you combine alllike terms in the polynomial before you multiply.

Examples

Find

Solution

Combine like terms in the polynomial and then multiply using the distributive property.

## Multiplying Polynomials

Use the distributive property to multiply polynomials. If you are multiplying two binomials, you can use a shortcut called the FOIL method.

To multiply two binomials, find the sum of the products of

 FOIL Method for Multiplying Two Binomials F the First termsO the Outer terms I the Inner terms L the Last terms

Example

Find (2x + 3)(4x - 1).