Lesson Video: Polynomial Functions Mathematics • Third Year of Preparatory School

Video Transcript

In this video, we’ll learn how to identify, write, and evaluate a one-variable polynomial function and state its degree and leading coefficient.

We read this as 𝑓 of 𝑥, where 𝑓 is the function of the single variable 𝑥. The degree, or order, of 𝑓 is 𝑛, which is the highest power of the variable 𝑥 and must be a nonnegative integer. And the coefficients 𝑎 sub 𝑖, where 𝑖 goes from zero to 𝑛, are real constants. Up to now, you’ll already have been working with some polynomial functions, perhaps without even realizing it. For example, the area of a square, which we can call 𝐴 of 𝑥, so 𝐴 is a function of 𝑥, is the square of its side length. And this is a polynomial function of degree two. A polynomial of degree two is called a quadratic, and this one, 𝐴 of 𝑥, has leading coefficient equal to one.

The volume of a cube, which we can call 𝑉 of 𝑥, is equal to the side length cubed and this is a polynomial of degree or order three. We call this a cubic polynomial, and again, in this case, its leading coefficient is equal to one. Other examples of polynomials include linear functions, which have degree one. So, 𝑛 is equal to one, and a linear function has the form 𝑓 of 𝑥 equals 𝑎 sub one times 𝑥 plus 𝑎 sub zero. And in the example shown, the leading coefficient, 𝑎 sub one, is equal to three and the constant, 𝑎 sub zero, is equal to seven. For a linear function, since anything to the power of one is itself, we can leave out the power 𝑛 equal to one of 𝑥.

It’s also worth pointing out that since the powers of 𝑥 range from zero to 𝑛, we are including 𝑥 to the power zero in polynomial functions. It’s just that since anything to the power of zero is equal to one and anything multiplied by one is itself, we don’t have to explicitly write 𝑥 to the power zero. A polynomial function of order or degree 𝑛 doesn’t necessarily include every nonnegative integer power of 𝑥 less than 𝑛. So, for example, 𝑔 of 𝑥 as shown has degree four but only includes 𝑥 to the powers four, two, and zero, not 𝑥 to the power three or one.

Now remember, the degree 𝑛 must be a nonnegative integer, that’s a positive whole number or zero. So functions like 𝑓 of 𝑥 is equal to the square root of 𝑥 are not polynomial functions. And that’s because the square root of an expression means the expression to the power of a half, so the power or exponent of 𝑥 is not a nonnegative integer. Similarly, functions like 𝑓 of 𝑥 equals one over 𝑥 plus two are not polynomial functions. Because one over an expression is that expression to the power negative one, which is a negative integer.

On the other hand, the two functions shown of degrees two and three, respectively, are polynomial functions. The function 𝑓 of 𝑥 equals two 𝑥 squared plus 11𝑥 minus one is another example of a quadratic function. And 𝑓 of 𝑥 equals four minus 𝑥 to the power three, or 𝑥 cubed, plus two 𝑥 squared is another example of a cubic function. In fact, we can have polynomials where 𝑛 is any nonnegative integer. So, our degree could be, for example, 42, or seven, as in the final two examples, respectively.

To make our definition of a polynomial function a little more formal, we define monomials, which are the building blocks of polynomials, as the product of constants and variables where the variables may have only nonnegative integer exponents. Consider the list of expressions (a) to (g). Let’s see which of these are monomials.

Expression (a) can be rewritten as 𝑥 to the power one. And since one is a nonnegative integer exponent, this is a monomial. Expression (b) consists of a variable 𝑡 to the power positive six, so this too is a monomial since the exponent six is a positive integer. Expression (c), on the other hand, can be rewritten as 𝑥 raised to the power one-third, which is not a nonnegative integer. So, this expression is not a monomial.

For (d), zero is actually a monomial, since zero can be written as zero times 𝑥, or any other power of 𝑥. In fact, as was indicated previously, any constant 𝑐 is a monomial, since 𝑐 can be written as 𝑐 times 𝑥 to the power zero. That’s 𝑐 times one. Now for expression (e), this is not a monomial since it contains more than one term, although it is actually the sum of monomials 𝑥 squared and one. Expression (f) is also not a monomial, because negative two, which is the exponent of 𝑦, is a negative integer.

And finally, expression (g) is a monomial. It’s a single term and every variable in that term is raised to a nonnegative integer exponent. We can rewrite this as shown. And the fact that the constant three over two is not an integer doesn’t matter, since it’s only the exponents of the variables that must be nonnegative integers. Note also that this is a multivariable monomial since there are three variables, 𝑥, 𝑦, and 𝑧.

So, expressions (a), (b), (d), and (g) are monomials.

We define a polynomial as an expression that is the sum of monomials, where each term is called a monomial term. A function that is polynomial is called a polynomial function. And we see that each term in our polynomial function 𝑓 of 𝑥 is a monomial. Let’s look at an example where we identify which functions are polynomial functions.

Which of the following is a polynomial function? Option (A) 𝑓 of 𝑥 equals the square root of 𝑥 plus four. Option (B) 𝑓 of 𝑥 equals 𝑥 raised to the power negative two plus two 𝑥 plus four. Option (C) 𝑓 of 𝑥 equals one over 𝑥. Option (D) 𝑓 of 𝑥 equals two times 𝑥 raised to the power negative two. Or option (E) 𝑓 of 𝑥 equals 𝑥 squared plus two 𝑥 plus four.

To answer this question, we recall that, by definition, every term of a single-variable polynomial function must be a monomial. That is a product of constants and a single variable with only nonnegative integer exponents. Let’s go through each option one by one to see if they match this definition.

First, we see that option (A) contains the term root 𝑥, which is equivalent to 𝑥 raised to the power of one-half. Since this is a noninteger power of the variable, option (A) cannot be a polynomial function. Now, if we consider option (B), this time the function contains a negative integer power of 𝑥, that is, negative two. So, option (B) cannot be a polynomial function. And in fact, option (D) contains the same power of 𝑥. So, we can discount option (D) for the same reason.

Now, let’s look at option (C). By the laws of exponents, we know that one over 𝑥 can be written as 𝑥 raised to the power negative one. And since this is 𝑥 to a negative integer, option (C) cannot be a polynomial function. This leaves option (E).

Going through each term in option (E), we see that, first, 𝑥 squared is the variable 𝑥 raised to a positive integer exponent. Two 𝑥 can be written as two times 𝑥 raised to the power one, so this term is the product of a constant, two, and the single variable 𝑥 raised to a positive integer exponent. And the final term, four, is a constant, which can be written as four times 𝑥 raised to the power zero. Since this term is a monomial, the function 𝑓 of 𝑥 equals 𝑥 squared plus two 𝑥 plus four is a sum of monomials. So, only option (E) is a polynomial function.

Now, let’s see how we can construct a polynomial function from given information about a real-world problem.

A bus service charges a fixed fee of five pounds and an additional two pounds for every bus stop passed. Write a polynomial function to represent the cost of a ride.

To construct a polynomial function representing the cost of a ride on the bus, we first need to extract the relevant information from the question. We’re told that there’s a fixed fee of five pounds. This five pounds will be a constant in our function. Next, we’re told that there is an additional fee of two pounds for every bus stop passed.

The number of bus stops passed is a variable quantity, so let’s call this 𝑥. And it’s worth noting that 𝑥 must be a positive integer since it represents the number of bus stops passed. This means that if we pass 𝑥 number of bus stops, we need to pay two 𝑥 pounds plus the fixed fee of five pounds. This is the total cost of the bus ride. And writing this as a function of 𝑥, we have 𝑓 of 𝑥 is equal to two 𝑥 plus five. Our answer is therefore 𝑓 of 𝑥 is equal to two 𝑥 plus five.

It’s worth just checking that this function is actually a polynomial, since the question explicitly asks for a polynomial function. To do this, we recall a couple of definitions. The first is a monomial. This is a product of constants and variables, where the exponents of the variables can only be nonnegative integers. The second definition is that of a polynomial function. This is a function that is a sum of monomials.

In our case, our first term is two 𝑥. Now, two 𝑥 is actually two times 𝑥 raised to the power one. So, we have the product of a constant, two, and 𝑥 raised to a positive integer exponent, one. This term is therefore a monomial. Our second term is the constant five. And this can be written as five times 𝑥 to the power zero, recalling that 𝑥 to the power zero is equal to one. So, the second term five is also a monomial, and our function 𝑓 of 𝑥 is the sum of monomials.

Hence, the cost of the bus journey, where 𝑥 is the number of bus stops passed, can be represented as the polynomial function 𝑓 of 𝑥 is equal to two 𝑥 plus five. Recall that to evaluate a function at a specific value of the variable, 𝑥, say 𝑥 is equal to 𝑎, we substitute 𝑥 equals 𝑎 into 𝑓 of 𝑥 wherever 𝑥 occurs and then evaluate the result. For example, if we’re asked to evaluate 𝑓 of 𝑥 equals seven 𝑥 cubed minus four 𝑥 squared plus three at 𝑥 equals two, wherever we have an 𝑥 in 𝑓 of 𝑥, we substitute the value 𝑥 equals two. And since two raised to the power three, or cubed, is eight, and two squared is four, this gives 56 minus 16 plus three, which is 43.

Let’s see another example of this.

If 𝑓 of 𝑥 is equal to negative eight 𝑥 squared minus three 𝑥 plus four, find 𝑓 of negative three.

We’re asked to find the value of 𝑓 of negative three. And we recall that this is function notation for the value of 𝑓 of 𝑥 when 𝑥 is equal to negative three. This means in our function 𝑓 of 𝑥, wherever we have an 𝑥, we substitute negative three. So, we have 𝑓 of negative three is equal to negative eight times negative three squared minus three times negative three plus four. That is negative eight times nine plus nine plus four, which evaluates to negative 59. Hence, 𝑓 of negative three is equal to negative 59.

Before moving on to some more examples, let’s remind ourselves of some of the terminology that will help us describe the type of polynomial function we’re working with. Remember, for a single-variable polynomial, the largest exponent of a variable in any nonzero term is called the degree or order of a polynomial. The term in a polynomial with the highest degree is called the leading term of the polynomial, and the constant factor of the leading term in a polynomial is called the leading coefficient. Let’s look at an example.

Find the degree and leading coefficient of the polynomial function 𝑓 of 𝑥 is equal to three 𝑥 to the fourth power plus two 𝑥 cubed plus five 𝑥 squared plus seven.

To answer this, we recall that for a single-variable polynomial function, the degree of the polynomial is the largest exponent of a variable in any nonzero term. To find the degree of the given polynomial function, we note that the only variable is 𝑥. And we can rewrite the final term to include 𝑥 to the power of zero, since 𝑥 to the power zero is equal to one. The variable then appears in each nonzero term. We can then see that 𝑥 has exponents four, three, two, and zero. And the largest of these exponents is equal to four. The degree of the function is therefore four.

We note further that the term in a polynomial with the highest degree is called the leading term of the polynomial. And in our case, the term with the highest degree is the term where the exponent of 𝑥 is four. That is three times 𝑥 to the power four. So, this is our leading term. But we’re not asked for the leading term of the polynomial function; we’re asked for the leading coefficient. That is the constant factor of the leading term, and that’s equal to three. Hence, the degree of the given polynomial function is four and its leading coefficient is three.

We can gain information about the shape and complexity of a polynomial from its degree. And we give specific names to some families of polynomials based on their degree. We saw some of these at the beginning of this video. A polynomial function of degree zero is called a constant function. A polynomial function of degree one is called a linear function. A polynomial function of degree two is called a quadratic function. One of degree three is called a cubic function. A polynomial function of degree four is called a quartic function. And one of degree five is a quintic function.

As we’ve seen, a constant function has the form 𝑓 of 𝑥 equals 𝑎 for some real number 𝑎. We can write this as 𝑎 times 𝑥 to the power zero, since 𝑥 to the power zero is equal to one. Hence, the degree is zero. It’s worth noting, however, that for the special case where 𝑎 is equal to zero, this is called the zero polynomial. And recalling the definition of degree, the largest exponent of a variable in any nonzero term, in this special function, every term is zero. Hence, we leave the degree of the zero polynomial as undefined. There are names for polynomials of degree higher than five, but we don’t commonly use these. Let’s look at an example of determining the type of a polynomial function.

Identify the name of the polynomial function 𝑓 of 𝑥 equals two 𝑥 squared plus four 𝑥 cubed plus three 𝑥 plus five.

Now we might be tempted to name our polynomial function something like Fred or Philomena. But that would be silly. Instead, we recall that we name polynomial functions based on their degree. That is, in a single-variable polynomial, the degree is the largest exponent of a variable in any nonzero term. We can rewrite the given function as shown so that every term is a product of a constant and a variable to a power. Hence, the final term is actually five 𝑥 to the power zero and the term before that is three 𝑥 to the power one.

We see now that the largest exponent of the variable 𝑥 is three in the second term. So, this is the degree of the polynomial. Finally, we recall that single-variable polynomials of degree three are called cubic functions. Hence, 𝑓 of 𝑥 is a cubic function.

Let’s now complete this video by recapping some of the important points we’ve covered.

First, a monomial is a product of constants and variables where the variables can have only nonnegative integer exponents. A polynomial is an expression that is a sum of monomials. A single-variable polynomial is a polynomial containing a single variable. The degree of a polynomial is the largest exponent of the variable in any nonzero term. The leading term of a polynomial is the term with the highest degree. The leading coefficient is the constant factor of the leading term.

And finally, certain types of polynomial functions have specific names based on their degree. A polynomial of degree zero is called a constant function. Degree one is a linear function. Degree two is a quadratic function. Degree three is a cubic. Degree four is called a quartic function. And a degree five polynomial is called a quintic function.