Question

if $x + 3$ is a factor of the polynomial $x^3 + 2x^2 - 5x + n$, what is the value of $n$?

choose the correct answer below.

a. $-3$

b. $-2$

c. $-6$

d. $3$

e. $6$

Explanation:

Step1: Recall Factor Theorem

If \(x + a\) is a factor of a polynomial \(P(x)\), then \(P(-a)=0\). Here, \(x + 3\) is a factor, so \(a = 3\), and we need to find \(P(-3)=0\) where \(P(x)=x^{3}+2x^{2}-5x + n\).

Step2: Substitute \(x=-3\) into \(P(x)\)

Substitute \(x=-3\) into \(P(x)\):

$$
LATEXBLOCK0
$$

Step3: Solve for \(n\)

Since \(P(-3) = 0\) (by Factor Theorem), we set \(6 + n=0\). Solving for \(n\), we get \(n=-6\).

Answer:

C. \(-6\)