Question

determine which of the following equations has roots -1, -5, and 4
a) ( x^3 + 29x - 26 = 0 )
b) ( x^3 - 2x^2 + 19x + 26 = 0 )
c) ( x^3 - 8x^2 - 11x - 20 = 0 )
d) ( x^3 + 2x^2 - 19x - 20 = 0 )

Explanation:

Step1: Recall Factor Theorem

If \( r \) is a root of a polynomial \( P(x) \), then \( (x - r) \) is a factor of \( P(x) \). For roots \( -1, -5, 4 \), the factors are \( (x + 1) \), \( (x + 5) \), \( (x - 4) \).

Step2: Multiply the Factors

First, multiply \( (x + 1)(x + 5) \):
\( (x + 1)(x + 5)=x^{2}+5x + x + 5=x^{2}+6x + 5 \).
Then multiply by \( (x - 4) \):
\( (x^{2}+6x + 5)(x - 4)=x^{3}-4x^{2}+6x^{2}-24x + 5x - 20 \).
Simplify: \( x^{3}+2x^{2}-19x - 20 \).

Answer:

D. \( x^{3}+2x^{2}-19x - 20 = 0 \)