Identify the name of the polynomial function 𝑓 of 𝑥 equals two 𝑥 squared plus four 𝑥 cubed plus three 𝑥 plus five.
So we’ve been given this polynomial function. This is the sum of monomial terms. And we’re asked to identify its name. So let’s begin by recalling the name of some of the polynomial functions we should know and their features. The first polynomial function we’re interested in has a degree of zero. We call this a constant function because it’s made up of a constant, for example, four. But what do we mean when we talk about the degree of a polynomial function? The degree of a polynomial function in fact is the highest exponent of each of the terms. If the highest exponent is zero then, we say that it is degree zero. And what about if it has a degree of one? Well, this is said to be a linear function. And an example of this is something like three 𝑥 or four 𝑦.
In each case, each of these terms has an exponent of one, so the degree of each function is one. Then there’s a quadratic function. The degree of this function is two. In other words, the highest exponent of any of its terms is two. For instance, take the function seven 𝑥 squared plus 𝑥. This contains terms with exponents two and one. So its highest exponent is two. If we think about, however, a function such as two 𝑥𝑦, we find the sum of the exponents to calculate the exponent of the expression. In this case, one plus one is two. Finally, a degree three function is known to be cubic, for instance, 𝑥 cubed minus two 𝑥𝑦 plus four. Since this function is degree three, its highest exponent is indeed three.
So with these four main functions in mind, let’s have a look in a little more detail at our function 𝑓 of 𝑥. Let’s begin at the very end of this function. The term five has an exponent of zero, since five is the same as five 𝑥 to the zeroth power. Five is a constant term, but 𝑓 of 𝑥 is not a constant function. Then, our term three 𝑥 has an exponent of one. Four 𝑥 cubed has an exponent of three. And finally, the term two 𝑥 squared has an exponent of two. We see that the highest exponent here is in fact three. This means the degree of 𝑓 of 𝑥 is three. And so the name of our polynomial function is cubic. It’s a cubic function.