Why aren't all polynomial functions of odd degree, odd functions? [closed]

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If it's an odd degree, why isn't it always an odd function? Can you please give me an example?

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2 Answers

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An odd-degree mononomial is an odd function. A polynomial is an odd function if every one of its trms is of odd degree. Thus $$x^{17} +5ox^9-35x$$ is an odd function. But the degree of a polynomial is defined to be the degree of the term of highest degree, the so-called leading term, so $$3x^3+5x^2$$ is of odd degree because its leading term has degree 3, but the presence of a term of even degree prevents the polynomial from being an odd function.

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This is because to make $f(x)$ odd, it must satisfy $f(x) = -f(-x)$. Now in polynomials, if $f(x)$ is odd then it must have all the powers of variable odd.

We can prove this by showing that other cases where polynomial has atleast 1 even power does not satisfy.

Eg:$f(x) = x + 1$

$f(-x) = -x + 1$

$f(x) ≠ -f(-x)$

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