Question

Use Descartes' Rule of Signs to find the number of possible positive real roots and the number of possible negative real roots for the function f(x) = x^4+ 2x^3-3x^2- 8x - 4.

a positive 1; negative 3 or 1


b. positive 1; negative 3 or 5


C. positive 3; negative 3 or 1


d. positive 3; negative 3 or 5

Answer 1

a.positive 1; negative 3 or 1

Explanation:

EDGE 2020

Answer 2

a positive 1; negative 3 or 1

Explanation:

To determine the number of positive roots, we have to determine the number of sign changes for f(x) = x⁴ + 2x³ - 3x² - 8x - 4.

The coefficients in f(x) are +1, +2, -3, -8, -4.

Since there is only one sign change from +2 to -3, we have 1 positive root.

To determine the number of negative roots, we have to determine the number of sign changes for f(-x) = (-x)⁴ + 2(-x)³ - 3(-x)² - 8(-x) - 4 = x⁴ - 2x³ - 3x² + 8x - 4

The coefficients in f(-x) are +1, -2, -3, +8, -4.

Since there is three sign change from +1 to -2, from -3 to +8, and from +8 to -4. So,we have 3 or 1 negative root, since the number of negative roots is equal to the number of sign changes or an even number less than the number of sign changes. So, 3 -2 = 1

So, the number roots are of positive 1; negative 3 or 1