Question

using descartes' rule of signs, we can tell that the polynomial p(x) = x5 − 6x4 2x3 − x2 7x − 9 has, from smallest to largest, , , or positive real zeros and negative real zeros.

Answer 1

The polynomial p(x) = x^5 - 6x^4 + 2x^3 - x^2 + 7x - 9 has either 2 or 0 positive real zeros and 1 negative real zero.

Step-by-step explanation:

The polynomial p(x) = x5 − 6x4 + 2x3 − x2 + 7x − 9 is given. To determine the number of positive and negative real zeros, we can use Descartes' Rule of Signs.

1. Count the number of sign changes in the coefficients as we move from left to right. In this case, we have 2 sign changes: from -6x4 to +2x3, and from -x2 to +7x. Therefore, there are either 2 or 0 positive real zeros.
2. Let's substitute -x for x in the polynomial and count the sign changes again. We now have 1 sign change: from -6(-x)4 to +2(-x)3. Therefore, there is 1 negative real zero.

So, using Descartes' Rule of Signs, the polynomial p(x) = x5 − 6x4 + 2x3 − x2 + 7x − 9 has either 2 or 0 positive real zeros and 1 negative real zero.