Which of the following expressions are monomials: a) 𝑥 squared, b) 𝑥 cubed 𝑦 to the power of four, c) 𝑥 to the power of negative three, d) one divided by 𝑦, and e) one plus 𝑥?
If we firstly consider the definition of a polynomial, this can be made up of constants, variables, and exponents. But it can never be divisible by a variable. This immediately rules out option d, one divided by 𝑦, and option c, 𝑥 to the negative three, as this is the same as one divided by 𝑥 cubed. Both of these options are being divided by a variable.
In the case of c, 𝑥 cubed; and in the case of d, the variable 𝑦. A monomial is a special type of polynomial. It is a polynomial with just one term. This also rules out option e, one plus 𝑥, as this has two terms.
Both option a, 𝑥 squared, and option b, 𝑥 cubed 𝑦 to the power of four, are polynomials with one term. Therefore, they are monomials. This means that two of the five expressions are monomials: a 𝑥 squared and b 𝑥 cubed 𝑦 to the power of four.