Polynomials "Bouncing" off the X-Intercept

Question

I was struggling with an explanation for why polynomials with even exponents bounce off the line Y=0, while ones with odd exponents go through. I'm assuming it has to do with signs, and how - times - is positive ect., but we've never had it explained fully to us in class. Why does it come back up after it hits?

Answer 1

For the record, not all polynomials of even degree "bounce" off the x-axis: try playing around with some 4th or 6th degree polynomials in Desmos (https://www.desmos.com/calculator)

The above, for example, is the graph of $f\left( x\right)=x^6+x^5-x^4+x^3-x^2-2x+1$

But I think I get the phenomenon you're trying to get at - why do polynomials of even degree have both "ends" of it go off to $-\infty$ or $+\infty$, when the ones of odd degree always have one going to $-\infty$ and the other to $+\infty$?

The answer mostly lies in signs, and how exponents handle them.

Consider, for example, $(-2)^n$, where $n$ is some positive whole number. Notice:

• $n$ is even:

  • $n = 2 \Rightarrow (-2)^n = 4$
  • $n = 4 \Rightarrow (-2)^n = 16$
  • $n = 6 \Rightarrow (-2)^n = 64$

• $n$ is odd:

  • $n = 1 \Rightarrow (-2)^n = -2$
  • $n = 3 \Rightarrow (-2)^n = -8$
  • $n = 5 \Rightarrow (-2)^n = -32$
  • $n = 7 \Rightarrow (-2)^n = -128$

Notice how the sign is handled: if $n$ is even, the sign is always positive, and if $n$ is odd, then the sign is negative (if the number we're looking at is negative).

(Minor nuance: these all assume they're regarding polynomials of positive leading coefficient. If the coefficient is negative, then the entire graph flips upside-down, basically, and thus things that go to $+\infty$ now to go $-\infty$, and vice versa, and the signs mentioned above would all flip.)

Further, if you ignore whether the number is positive or negative, notice that the numbers get bigger the further away from the origin you go on the $x$ axis. (Mathematically, we say that the numbers are increasing in magnitude the further from the $x$ axis you go.) In polynomials, the bigger the input, the bigger the output - that's just how things work, and it becomes clear when you consider every term of the polynomial has some $x^n$ thing in there. So if $x$ is a really really big number, then $x^n$ is a really REALLY big number.

These two things basically describe a rough idea of why polynomials of even and odd degree act like they do: terms of even degree like to turn the signs positive while odd degree terms like to keep the sign, and the further out you go the bigger in size the terms get (though depending on the sign, they start going off to $-\infty$ or $+\infty$, depending).