Question

1. what is the value of ( f(-2) ) of the polynomial ( f(x)=-x^{3}+x + 1 )? what does this mean about the linear binomial ( (x + 2) )?

( \bigcirc f(-2)=7 ); this means that ( (x + 2) ) is not a factor of ( f(x)=-x^{3}+x + 1 ).
( \bigcirc f(-2)=-5 ); this means that ( (x + 2) ) is not a factor of ( f(x)=-x^{3}+x + 1 ).
( \bigcirc f(-2)=0 ); this means that ( (x + 2) ) is a factor of ( f(x)=-x^{3}+x + 1 ).
( \bigcirc f(-2)=5 ); this means that ( (x + 2) ) is not a factor of ( f(x)=-x^{3}+x + 1 ).

Explanation:

Step1: Substitute \(x = - 2\) into the polynomial

Given \(f(x)=-x^{3}+x + 1\), substitute \(x=-2\):

$$
LATEXBLOCK0
$$

Step2: Calculate \((-2)^{3}\)

\((-2)^{3}=(-2)\times(-2)\times(-2)=-8\), so \(f(-2)=-(-8)-2 + 1\)

Step3: Simplify the expression

\(-(-8)=8\), then \(f(-2)=8-2 + 1=7\)

According to the factor theorem, if \(f(a)=0\), then \((x - a)\) is a factor of \(f(x)\). Here \(a=-2\), and \(f(-2)=7
eq0\), so \((x + 2)\) (since \(x-(-2)=x + 2\)) is not a factor of \(f(x)\)

Answer:

A. \(f(-2)=7\); this means that \((x + 2)\) is not a factor of \(f(x)=-x^{3}+x + 1\)