Question

question
which of the following functions has a factor of ( x + 1 )?
answer
( f(x) = x^3 + 0x^2 - 13x + 12 )
( f(x) = x^3 + 2x^2 - 11x - 12 )
( f(x) = x^3 + 2x^2 - 11x - 11 )
( f(x) = x^3 + 0x^2 - 13x + 11 )

Explanation:

Step1: Recall Factor Theorem

The Factor Theorem states that if \( x + a \) is a factor of a polynomial \( f(x) \), then \( f(-a) = 0 \). For the factor \( x + 1 \), we have \( a = 1 \), so we need to check \( f(-1) = 0 \) for each function.

Step2: Check \( f(x)=x^3 + 0x^2 - 13x + 12 \)

Substitute \( x = -1 \):
\( f(-1)=(-1)^3 + 0(-1)^2 - 13(-1) + 12 = -1 + 0 + 13 + 12 = 24
eq 0 \).

Step3: Check \( f(x)=x^3 + 2x^2 - 11x - 12 \)

Substitute \( x = -1 \):
\( f(-1)=(-1)^3 + 2(-1)^2 - 11(-1) - 12 = -1 + 2 + 11 - 12 = 0 \).

Step4: Check \( f(x)=x^3 + 2x^2 - 11x - 11 \)

Substitute \( x = -1 \):
\( f(-1)=(-1)^3 + 2(-1)^2 - 11(-1) - 11 = -1 + 2 + 11 - 11 = 1
eq 0 \).

Step5: Check \( f(x)=x^3 + 0x^2 - 13x + 11 \)

Substitute \( x = -1 \):
\( f(-1)=(-1)^3 + 0(-1)^2 - 13(-1) + 11 = -1 + 0 + 13 + 11 = 23
eq 0 \).

Answer:

\( f(x) = x^3 + 2x^2 - 11x - 12 \) (the third option in the list)