Exercise 21 Page 340

To find the least possible degree of a polynomial function, there are two things we should consider.

1. The number of turning points plus one.
2. The number of real zeros.

Whichever of these two numbers is larger is the smallest possible degree.

Turning Points

Let k be the number of turning points and n be the degree of the given function. We know, that the polynomial function has at most n-1 turning points. k ≤ n - 1 ⇔ k + 1 ≤ n The smallest possible degree of a polynomial function is given by the number of turning points plus one. k turning points ⇒ k+1 degrees From the previous exercise, we know that this function has 3 turning points. Therefore, 3+1=4 is one possibility for the smallest possible degree.

Real Zeros

Now, let's consider the number of real zeros. n zeros ⇒ n degrees We know from the previous exercise that the function has 4 real zeros. Therefore, the other possibility for the smallest possible degree of our function is 4.

Smallest Possible Degree

As we can see, the number of turning points plus one and the number of real zeros are the same number, 4. This means that the smallest possible degree of our polynomial is 4.