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How to tell whether a function is a polynomial

Look at every exponent

A polynomial in $x$ is a finite sum of constant multiples of nonnegative integer powers of $x$ . A variable in a denominator has a negative exponent; a square root of $x$ has exponent $1/2$ . Neither is allowed.

Problem

Which are polynomial functions? (a) $v (x) = 7 x^{6}$ ; (b) $h (x) = - 1/ x^{3}$ ; (c) $u (x) = 3 x - 9 x^{5}$ ; (d) $f (x) = x^{9} - 7 + 3 x^{- 6}$ .

Answer

Only (a) is a polynomial function.

A negative coefficient is fine; a negative or fractional exponent of x is not.

Step-by-step solution

1. Apply the definition

Every exponent of $x$ in a polynomial must be a nonnegative integer. Constants are allowed.

2. Classify the four functions

(a) $7 x^{6}$ has exponent 6, so it is a polynomial. (b) $- 1/ x^{3} = - x^{- 3}$ has a negative exponent. (c) $x = x^{1/2}$ has a fractional exponent. (d) The term $3 x^{- 6}$ has a negative exponent. Thus (b), (c) and (d) are not polynomials.

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