Question

Determine the real zeros and state the multiplicity of any repeated zeros of

f(x)=−x⁵+5x⁴−9x³ +7x²−2x.
A) One real zero with multiplicity 3
B) Two real zeros with multiplicity 2
C) One real zero with multiplicity 4
D) Three real zeros with multiplicity 1

Answer 1

The function has the factor x, indicating a real zero at x = 0 with multiplicity of 1. Without further factoring or numerical methods, the exact real zeros and multiplicities of the quartic polynomial in f(x) cannot be determined, making it impossible to accurately choose from the given options.

Step-by-step explanation:

To determine the real zeros and their multiplicity for the function f(x) = -x⁵ + 5x⁴ - 9x³ + 7x² - 2x, we need to factor the function as much as possible. In this function, we can immediately see that there is a common factor of x in each term. We factor it out:

f(x) = x(-x⁴ + 5x³ - 9x² + 7x - 2)

Setting the function equal to zero:

0 = x(-x⁴ + 5x³ - 9x² + 7x - 2)

The zeros of the function are the solutions to this equation. We already know one solution is x = 0 with multiplicity 1 because x is a factor of the polynomial. To find the other zeros, we need to investigate the quartic polynomial:

-x⁴ + 5x³ - 9x² + 7x - 2