Questions: What are all of the real roots of the following polynomial? f(x)=x^5+5x^4-5x^3-25x^2+4x+20 A. -5,-2, and 2 B. -5,-2,-1,1, and 2 C. -3,-2,-1,1, and 2 D. -2,-1,1, and 2

Transcript text: What are all of the real roots of the following polynomial? \[ f(x)=x^{5}+5 x^{4}-5 x^{3}-25 x^{2}+4 x+20 \] A. $-5,-2$, and 2 B. $-5,-2,-1,1$, and 2 C. $-3,-2,-1,1$, and 2 D. $-2,-1,1$, and 2

Solution

Solution Steps

To find the real roots of the polynomial $ f(x) = x^5 + 5x^4 - 5x^3 - 25x^2 + 4x + 20 $, we can use numerical methods or root-finding algorithms. One common approach is to use Python's numpy library to find the roots of the polynomial, which will include both real and complex roots. We then filter out the real roots.

Step 1: Identify the Polynomial Roots

The polynomial given is:

\[ f(x) = x^5 + 5x^4 - 5x^3 - 25x^2 + 4x + 20 \]

The roots of this polynomial are calculated as:

\[ -5, 2, 1, -2, -1 \]

Step 2: Verify the Real Roots

The calculated roots are all real numbers. We verify them by checking if they satisfy the polynomial equation $ f(x) = 0 $.