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At a Glance - Degrees of a Polynomial

Each term in a polynomial has what's called a degree, or a value based on the exponent attached to its variable. The degree of 9x2 is 2, for example. You may be unfamiliar with a degree of 2 unless you've ever been to Fairbanks, Alaska, in the middle of January.

We usually write the terms of a polynomial in descending order (greatest to least) according to the degree of each term. Exactly like a shuttle launch countdown. For example, we would write 2x2 + x instead of x + 2x2.


7 0

The degree of the constant 7 is zero, since 7 = 7x0. The degree of any other non-zero constant is also zero. However, the degree of the term 0 is undefined. You might want to get on that, Webster's.

Since 0 = 0x = 0x1 = 0x2 = 0x3 (all of these are the same as 0), any degree would work, and there's no obvious way to decide which one to pick. Mathematicians hate not knowing what to pick, so that's why they say it's "undefined." You should see them in a grocery store looking over grapefruit to see which is the most ripe. "Undefined...undefined...undefined...undefined..."

There's also a degree of each polynomial. You can figure out the degree of a polynomial if you haven't forgotten which numbers are bigger than each other. If all else fails, count them off on your fingers and hope you never run into anything bigger than 10.

Memorize this: the degree of a polynomial is the largest degree of any one term in the entire polynomial. While we usually write polynomials with the largest degree term first, it's a good idea to look at the degrees of all the terms, in case some impish degree sprite came along and mixed them up to make our lives miserable.

More Samples

5x3 + 6x + 93
x23 + x6 + 4x – 223
4x6 + 3x5 + 2x36
7 0
4x + 5x22

The degree-0 term of a polynomial is also called the constant term of the polynomial—the number sitting all by itself, usually at the end of the polynomial. Who knows, maybe it couldn't find its deodorant this morning.

If a polynomial doesn't seem to have a constant term, as in 3x2 + 4x, we say its constant term is 0 because we can write "+ 0" at the end of any expression without changing the value of the expression. If you're ever asked to pay $10 + 0 for something, remember that it doesn't cost you anything extra before you decide to get all in a huff about it.

Example 1

In the polynomial 5x2 + 3x + 4, the constant term is what?

Example 2

The constant term of the polynomial x5 + 5x2 – 7 is what?

Example 3

The polynomial 5x2 + 2x has a constant term of what?

Exercise 1

Find the degree and constant term of the polynomial x4 + 7x3 – 2x.

Exercise 2

Find the degree and constant term of the polynomial 3x20x10 + x9 – 5.

Exercise 3

Find the degree and constant term of the polynomial x15 + 8.

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