Use Descartes's Rule of Signs to determine the possible numbers of positive and neg- ative zeros of the function. $5 x ^ { 5 } - 10 x$
A) One positive zero, one negative zero
B) Two negative zeros or no positive zero
C) One negative zero
D) One negative zero, four negative zeros
E) One negative zero or three positive zeros

Question

Quizplus · Accepted Answer

To apply Descartes's Rule of Signs, we need to first arrange the polynomial in descending order:

$$5x^5 - 10x$$

The number of possible positive zeros is given by the number of sign changes in the coefficients of the polynomial or less than that by an even integer. There is one sign change, so there can be at most one positive zero.

The number of possible negative zeros is given by the number of sign changes in the coefficients of either $f(x)$ or $f(-x)$, so let's compute $f(-x)$:

$$5(-x)^5 - 10(-x) = -5x^5 + 10x$$

There are no sign changes between the coefficients of $f(x)$ and $f(-x)$, so there can be at most one negative zero.

Hence, the possible numbers of positive and negative zeros are one positive and one negative zero. Therefore, the answer is A).

Use Descartes's Rule of Signs to Determine the Possible Numbers 5x5−10x5 x ^ { 5 } - 10 x5x5−10x

Use Descartes's Rule of Signs to Determine the Possible Numbers $5 x ^ { 5 } - 10 x$