Which of the following is not a polynomial function? Is it (A) 𝑓 of 𝑥 equals one over 𝑥 plus two? Is it (B) 𝑓 of 𝑥 equals two? (C) 𝑓 of 𝑥 is 𝑥 to the fourth power minus two 𝑥 cubed plus two. (D) 𝑓 of 𝑥 equals 𝑥 squared minus four. Or (E) 𝑓 of 𝑥 is equal to the square root of two 𝑥 plus three.
Let’s begin by reminding ourselves what we mean when we talk about a polynomial function. A polynomial function is one that’s made up as a sum of monomial terms. So, what’s a monomial term? Well, a monomial term is a single term made up of a constant and a variable, and that variable will have a nonnegative integer exponent. So, for instance, three 𝑥𝑦 squared is a monomial. It’s the product of a constant three and some variables 𝑥 and 𝑦, and their exponents are one and two, respectively. Four 𝑥 to the power of negative seven, however, is not a monomial, and that’s because the exponent of the variable is negative seven.
So, let’s begin by identifying which of our functions are polynomials, are a sum of monomials. Well, let’s begin with (B) 𝑓 of 𝑥 equals two. This is the same as two 𝑥 to the zeroth power. Zero is nonnegative and it’s an integer. And so, we have a term made up of a constant a, variable, and a nonnegative integer exponent. So, in fact, (B) is a polynomial function.
So, what about option (C)? Well, we have 𝑥 to the fourth power. That’s a variable raised to a nonnegative integer exponent. We have negative two 𝑥 cubed, a constant times 𝑥 raised to another nonnegative integer exponent, and we just saw that two itself is indeed a monomial term. So, (C) is the sum of three monomials and it must be a polynomial function. In a similar way, (D) is also a polynomial function. It’s the sum of two monomials, 𝑥 squared and negative four.
So, what about option (E)? Well, we know that three is a constant. It’s three 𝑥 to the zeroth power, which is a monomial. But what about the square root of two times 𝑥? Well, the square root of two is in fact a constant, so we’re multiplying a constant by a variable, 𝑥 to the first power. This means root two 𝑥 is a monomial. And so, 𝑓 of 𝑥 is the sum of two monomials; it’s a polynomial.
And so, the answer must be option (A), but let’s double-check. We could rewrite 𝑓 of 𝑥 as 𝑥 plus two to the power of negative one. Now, we can’t distribute these parentheses. And so, in fact, we have a function which is a sum of a pair of monomials, but that itself is raised to a negative integer exponent. And so, 𝑓 of 𝑥 equals one over 𝑥 plus two cannot be a polynomial. And so, the answer is (A). (A) is not a polynomial function.