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What Are Degrees of Polynomials?

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Polynomials are expressions that are often used to model real world phenomena. Let's look at a characteristic of polynomials called the degree of a polynomial. We will see how to find the degree of a polynomial and why this information is useful.

Polynomials

Consider the following scenario. You are standing out on your porch with a ball in your hand. You throw the ball in the air in such a way that the height of the ball can be modeled by the following expression

-4.9x 2 + 100x + 5

where x is the number of seconds that pass after the ball leaves your hand.


Throwing a Ball From a Porch
degpoly1


In mathematics, this expression is called a polynomial. A polynomial is a mathematical expression containing terms made up of numbers, variables (the unknowns), and powers of variables combined together by addition or subtraction. The general form of a polynomial is shown.


degpoly2


We call the ai's coefficients, and the number we raise the variables to are called exponents. It is important to recognize that the exponent of the term x is 1, because x 1 = x, and the exponent of a constant term c is 0, because cx 0 = c*1 = c. Some more examples of polynomials are as follows:

  • 3x 4 - x 7 + 2x 5 + 5x - 1
  • x + 4
  • 8x 3 + 2x 2 - x + 7

Degree of a Polynomial

Let's talk about a certain characteristic of polynomials. This characteristic is called the degree of a polynomial. No, a polynomial does not have a temperature that can be measured in degrees. The degree of a polynomial is the greatest of all the exponents in the polynomial.

For example, let's go back to our polynomial that models the height of the ball that you threw off your porch. That is, -4.9x 2 + 100x + 5. We see that the exponent of the first term is 2, the exponent of the second term is 1, and the exponent of the third term is 0. Thus, the greatest exponent is 2, so the degree of the polynomial is 2. That's not so hard, is it?

This may leave you wondering why we would ever need to know the degree of a polynomial. Well, depending on the polynomial, the degree can tell us quite a bit about the polynomial and its characteristics.

For instance, we just found the degree of the polynomial -4.9x 2 + 100x + 5 to be 2. This tells us that the polynomial is a quadratic polynomial. It also tells us that the graph of this polynomial is a parabola, so it lets us know the general shape of the graph. The degree, along with the sign of the lead coefficient (or the coefficient in front of the variable with the highest exponent), tells us that the parabola will be pointing downward and that both ends of the parabola will go off in the downward direction.

In terms of the ball, all this information tells us that the ball starts off in an upward direction then peaks and begins to fall downward. The degree gives us a general idea of the trajectory of the ball. All of this information from the degree, and that just touches the surface! We see that though the degree of a polynomial is fairly simple to find, it is an extremely useful piece of information when analyzing a polynomial expression.

More examples

Let's get in a little more practice by finding the degrees of each of the polynomials given in the examples of polynomials above. We can start with the first one.

3x 4 - x 7 + 2x 5 + 5x - 1.

The first term has exponent 4, the second has exponent 7, the third has exponent 5, the fourth has exponent 1, and the last term has exponent 0. The highest of all these exponents is 7, so the degree of the polynomial is 7.

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