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.
![](polynomials-3-gifs/pic3.GIF)
Adding and Subtracting Polynomials
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 ![](polynomials-3-gifs/pic4.GIF)
Arrange like terms in column form and add. Follow the rules
for adding signed numbers.
![](polynomials-3-gifs/pic5.GIF)
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 ![](polynomials-3-gifs/pic6.GIF)
Solution
Combine like terms in the polynomial and then multiply using
the distributive property.
![](polynomials-3-gifs/pic7.GIF)
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 terms O the Outer terms
I the Inner terms
L the Last terms
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Example
Find (2x + 3)(4x - 1).
![](polynomials-3-gifs/pic8.GIF)
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