Video Transcript
Which of the following expressions is a polynomial? Is it option (A) the square root of 𝑥 plus five. Option (B) 𝑥 to the power of one-half plus five 𝑥. Is it option (C) 𝑥 squared plus five 𝑥 plus six? Option (D) 𝑥 to the power of negative one plus one. Or is it option (E) one over 𝑥 plus six 𝑥?
In this question, we’re given a list of five expressions involving a variable 𝑥. And we need to determine which of these expressions is a polynomial. So to answer this question, let’s start by recalling what we mean by a polynomial expression. We can recall a polynomial expression is one which is the sum of monomial expressions. And a monomial is the product of constants and variables, where we need the variables to be raised to nonnegative integer exponents. Therefore, we can answer this question by determining which of the five given expressions satisfies these properties.
Let’s start with the first expression. That’s the square root of 𝑥 plus five. We can see that this is not a polynomial expression. There’s no way we can remove the square root from this expression. And we know taking the square root of a number is the same as raising it to an exponent of one-half. And one-half is not an integer exponent anyway. So the expression in option (A) is not a polynomial.
We can see something very similar is true in the expression in option (B). The first term in this expression is 𝑥 raised to the power of one-half. This means the first term in this expression has a variable raised to a noninteger exponent. Therefore, the first term in this expression is not a monomial. So the expression in option (B) is not the sum of monomials. So, in turn, it is not a polynomial. The answer is not option (B).
Let’s now move on to the expression in option (C). Let’s check whether each of the terms is a monomial. In the first term, we have 𝑥 squared. This is a variable raised to the exponent of two, which is a nonnegative integer. So the first term is a monomial. In our second term, we have five multiplied by 𝑥. Remember, we can rewrite 𝑥 as 𝑥 to the first power, since raising a number to the power of one doesn’t change its value. So our second term is the constant five multiplied by the variable 𝑥 raised to the first power. This is the product of a constant and a variable where the variable is raised to a nonnegative integer exponent. This is a monomial.
Finally, there’s a few different ways of seeing that six is also a monomial. One way of doing this is to rewrite six as six times 𝑥 to the zeroth power. Now, this third term is the constant six multiplied by the variable 𝑥 raised to the zeroth power. This is a monomial expression. Therefore, the expression in option (C) is the sum of monomial terms, which means it is a polynomial expression.
We could stop here. However, for due diligence, let’s check the other two expressions we’re given. In the expression in option (D), we can see the first term is 𝑥 to the power of negative one. And this is a variable raised to a negative exponent. This means the first term is not a monomial. So the expression is not a polynomial. Therefore, the answer is not option (D).
We can in fact see something very similar is true in the expression in option (E). By using our laws of exponents, we know one divided by 𝑥 can be rewritten as 𝑥 to the power of negative one. So, once again, we have a variable raised to a negative exponent. This means the expression in option (E) is not a polynomial.
Therefore, we were able to show, of the five given expressions, only the expression in option (C), 𝑥 squared plus five 𝑥 plus six, is a polynomial.
