Stanley analyzes the table of values for a polynomial function. He has determined through applying Descartes’ rule of sign that there is 1 negative real zero, and either 1 or 3 positive real zeros. He has also determined that the upper bound of the function is 3.
X , f(x)
-2, -105
-1, 0
0, 5
1, -2
2, 3
3, -20
4, -175

a. What is the degree of the polynomial function? How can you tell?

b. How many positive real zeros does the function have? How can you tell?

a. The total number of real and complex zeros is equal to the degree of the polynomial. That total is (1 negative real) + (3 positive real/complex) = 4 total zeros. The degree of the polynomial is 4.

The even degree is confirmed by the answer to part b, and by the end-behavior shown in the table, which has a tendency to -∞ for |x|→∞.

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b. The intermediate value theorem tells you there will be zeros in the intervals (0, 1), (1, 2), and (2, 3) according to the values in the table. (The function changes sign in those intervals.) Thus there are 3 positive real zeros.

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Additional comment

Stanley cannot tell anything about Descartes' rule of signs by analyzing the table of function values. To use that rule, he must have terms of the polynomial. If he has those terms, he already knows the degree of the polynomial.