Chapter 9: Adding and Subtracting with Variables 117
Polynomials
There’s a particular group of expressions, which you’ll hear about in algebra, that are called
polynomials. The prefix poly means “many,” so it would seem that they would be expressions with
many terms, and some of them are. But the name polynomial is applied to any expression that fits a
particular pattern, even those with only one term.
The pattern is easier to show than to describe, but let’s try. Polynomials are expressions that are
made by adding terms that are the product of a numerical coefficient and a power of a certain
variable. For example, 8x^5 , -6x^3 , x^2 , and 2x are all terms that are the product of a numerical
coefficient and a power of the variable x. You don’t see the numerical coefficient in x^2 because
it’s 1, and we don’t usually show that. Constants can also be part of a polynomial because we can
say a constant, like 3, is 3x^0.
Each of these terms could be called a polynomial all by itself. It would be a one-term polynomial,
also called a monomial. Mono is the prefix that means “one.” You can add monomials with the
same variable to make a more complex polynomial. For example, 8x^5 + -6x^3 + x^2 + 2x + 3 is
a polynomial. If you add two monomials, like 5y^3 + 2y, you make a binomial. If you add three
monomials, like t^2 + 3t + 1, that’s a trinomial. Monomials, binomials, and trinomials are all types
of polynomials.
The degree of a monomial is the exponent on the variable. 8x^5 is fifth degree, x^2 is second degree,
and 2x is first degree. You don’t see an exponent of 1, but that’s what 2x really means: 2x^1.
Constants are degree zero, because we’re thinking of them as a constant times x^0.
MATH TRAP
Be careful to look at all the terms of a polynomial before you decide on its degree.
Don’t just jump at the term that happens to be written first.
The degree of a polynomial is the highest degree of all its monomials. The polynomial
8 x^5 + -6x^3 + x^2 + 2x + 3 is a fifth degree polynomial, and t^2 – 7t + 4 is a second degree poly-
nomial. For the polynomial 5t^3 -7t + 8t^4 -2t^2 + 5, you need to be careful to look at the whole
polynomial. The degree of the polynomial is 4, not 3. The highest degree term is in the middle,
not the beginning, of the polynomial.
If there is more than one variable in a term, the degree of the term is the sum of the degrees of
each variable. The term xy is degree 2, one for x and one for y. The term 3x^4 y^2 is degree 6.
