A polynomial function of degree n has at most_____ real zeros and at most_________turning points.

Answer 1

Hi Doug,

Thank you for your question.

--

Here is an in-depth video answer explanation with visuals : https://youtu.be/YzmfHMB4DJE

Simple answer: A polynomial function of degree n has at most n real zeros and at most n-1 turning points.

--

Explanation:

Remember the following.

1 ) The 'degree' of a polynomial is the value of its highest exponent. So x to the fifth power has a degree of 5. X squared has a degree of 2. X cubed has a degree of three. (x)(x)(x)(x) has a degree of four since it is equal to x to the 4th power.

2 ) Also, recall that a 'real zero' is a place where the polynomial will touch or cross the x-axis. This is also the place where each factor of a polynomial will equal 0. So if (x-2)(x+3) = 0, we have two 'real zeros' at x = 2 and x = -3.

Because of this last fact, we can keep multiplying factors which will give us additional real zeros.

(x) ----> one zero

(x-2)(x+3) -----> two real zeros

(x-2)(x+3)(x-5) ----> three real zeros

(x-2)(x+3)(x-5)(x+8) ----> four real zeros

Because when you set these polynomials equal to zero, they give real-number solutions (in this case, simple integers like 2, -3, 5, and -8).

So that is the real solution part.

The turning points is much easier to see in the video, but it makes sense if you think: "if something is going up, it needs to come back down to touch the x-axis" - which would be a turning point. Similarly, "if something is going down, it needs to come back up to touch the x-axis again." See the video for an explanation of that.

Let me know if this helped you.