Example 10 from Algebra and Trigonometry, 5.2 Power Functions and Polynomial Functions

Determining the Number of Intercepts and Turning Points of a Polynomial

Without graphing the function, determine the local behavior of the function by finding the maximum number of x-intercepts and turning points for $f (x) = - 3 x^{10} + 4 x^{7} - x^{4} + 2 x^{3} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The polynomial has a degree of $10,$ so there are at most 10 x-intercepts and at most 9 turning points.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Without graphing the function, determine the maximum number of x-intercepts and turning points for $f (x) = 108 - 13 x^{9} - 8 x^{4} + 14 x^{12} + 2 x^{3} .$

How did it go?

Solution

Your turn

Solution

Your turn