Video: Stating the Parity of a Polynomial Function

Is the function 𝑓(𝑥) = 8𝑥³ + 8𝑥, ∀ 𝑥 ∈ ℝ even, odd, or neither even nor odd?

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Video Transcript

Is the function 𝑓 of 𝑥 equals eight 𝑥 cubed plus eight 𝑥, for all real values of 𝑥, even, odd, or neither even nor odd?

We recall that any real-valued function is even if 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥. A function is odd, on the other hand, if 𝑓 of negative 𝑥 is equal to the negative of 𝑓 of 𝑥. In order to work out whether this function, eight 𝑥 cubed plus eight 𝑥, is even or odd, we need to find an expression for 𝑓 of negative 𝑥. We do this by replacing all our values of 𝑥 in the initial function with negative 𝑥.

Cubing negative 𝑥 means multiplying negative 𝑥 by negative 𝑥 by negative 𝑥. This is equal to negative 𝑥 cubed. This means that eight multiplied by negative 𝑥 cubed is equal to negative eight 𝑥 cubed. Multiplying eight by negative 𝑥 gives us negative eight 𝑥, as multiplying a positive by a negative gives a negative term. As both these terms are negative, we can factor or factorize out negative one. This can be rewritten as negative eight 𝑥 cubed plus eight 𝑥.

As the part inside the parentheses or bracket, eight 𝑥 cubed plus eight 𝑥, is equal to our initial function 𝑓 of 𝑥, then 𝑓 of negative 𝑥 is equal to negative 𝑓 of 𝑥. We can therefore conclude that the function eight 𝑥 cubed plus eight 𝑥, for all real values of 𝑥, is odd.

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