Question 1166523: The graphs of polynomial functions of degree n may have ______ to ____ zeros when n is even, and ____ to ____ when n is odd.
The graphs of polynomials functions of degree n have ____ to _____ hills and valleys when n is even and _____ to ______hills and valleys when n is odd.
thanks
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Any polynomial of degree n has n roots.
The graph of a polynomial function has a zero for each root which is real.
Non-real roots come in pairs.
Therefore, the graph of a polynomial of even degree can have no zeros, but the graph of a polynomial of odd degree must have at least one.
And it should be obvious that if all the roots are distinct real numbers, the graph of any polynomial function of degree n can have a maximum of n zeros.
So the answers to the first question are 0 to n for even degree functions and 1 to n for odd degree functions.
I'm not sure how you count "hills and valleys". Perhaps each time the graph "changes direction" in your terminology that creates a hill (function changes from increasing to decreasing) or a valley (changes from decreasing to increasing).
In that case, any function of ANY degree -- even or odd -- can have at most (n-1) hills and valleys.
And the graphs of y=ax^n for any degree n are all monotonic (always increasing or always decreasing, so no hills or valleys).
So the answers to the second question are 0 to (n-1) for both even and odd degree polynomials.
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