Polynomials

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

Examples

Consider the expression .

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

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

B What is the degree of the polynomial?

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

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

Adding and Subtracting Polynomials

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

Examples

Find each sum or difference.

A

Arrange like terms in column form and add. Follow the rulesfor adding signed numbers.

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

Find the additive inverse of - x + 2y. Then group the liketerms 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 amonomial. Youmay find it easier to multiply a polynomial by amonomial if you combine alllike terms in the polynomial beforeyou multiply.

Examples

Find

Solution

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

Multiplying Polynomials

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

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

Example

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