Video: Stating the Parity of a Polynomial Function

Is the function 𝑓(π‘₯) = βˆ’π‘₯⁡ βˆ’ 3π‘₯Β³ βˆ’ 8π‘₯, βˆ€ π‘₯ ∈ ℝ even, odd, or neither even nor odd?

03:03

Video Transcript

Is the function 𝑓 of π‘₯ equals negative π‘₯ to the power of five minus three π‘₯ cubed minus eight π‘₯, for all real values of π‘₯, even, odd, or neither even nor odd?

We recall that any real valid function is even if 𝑓 of negative π‘₯ is equal to 𝑓 of π‘₯. A function is odd, on the other hand, if 𝑓 of negative π‘₯ is equal to negative 𝑓 of π‘₯. In order to work out if our function negative π‘₯ to the power of five minus three π‘₯ cubed minus eight π‘₯ is even or odd, we need to find an expression for 𝑓 of negative π‘₯. We do this by replacing every π‘₯ term in our initial function by negative π‘₯.

We know that multiplying two negative terms together gives a positive answer. This means that squaring negative π‘₯ gives us π‘₯ squared. Cubing negative π‘₯ gives us negative π‘₯ cubed, as we’re multiplying a positive by a negative. This pattern continues such that negative π‘₯ to the power of four is π‘₯ to the power of four and negative π‘₯ to the power of five is equal to negative π‘₯ to the power of five.

When raising negative π‘₯ to any power, if the power is even, our answer will be positive, whereas if the power is odd, the answer will be negative. This means that the first term in our expression is negative negative π‘₯ to the power of five. The two negatives become a positive. So we are left with π‘₯ to the power of five.

Our second term becomes negative three multiplied by negative π‘₯ cubed. This is equal to three π‘₯ cubed. Finally, negative eight multiplied by negative π‘₯ is equal to eight π‘₯. The function 𝑓 of negative π‘₯ is equal to π‘₯ to the power of five plus three π‘₯ cubed plus eight π‘₯.

We notice that all three terms have the opposite sign from 𝑓 of π‘₯. We could therefore factor or factorize out negative one. This could be written as negative negative π‘₯ to the power of five minus three π‘₯ cubed minus eight π‘₯. The expression that is inside the parentheses or brackets is the same as 𝑓 of π‘₯.

We can therefore conclude that 𝑓 of negative π‘₯ is equal to negative 𝑓 of π‘₯. This means that the function is odd for all real values of π‘₯.

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