Question

applying remainder theorem:

1. khloe thinks that ( f(-2) ) is a zero of ( f(x) = 3x^3 + 5x^2 - x + 1 ). describe how you could use the remainder theorem to check to see if khloe is correct. is she correct?

Explanation:

Step1: Recall Remainder Theorem

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). If \( f(a) = 0 \), then \( x - a \) is a factor of \( f(x) \), and \( a \) is a zero of \( f(x) \). Here, we need to check \( f(-2) \) for the polynomial \( f(x)=3x^{3}+5x^{2}-x + 1 \).

Step2: Substitute \( x=-2 \) into \( f(x) \)

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

$$\begin{align*} f(-2)&=3(-2)^{3}+5(-2)^{2}-(-2)+1\\ &=3(-8)+5(4)+2 + 1\\ &=-24 + 20+2 + 1\\ &=(-24+20)+(2 + 1)\\ &=-4+3\\ &=-1 \end{align*}$$

\]

Answer:

Since \( f(-2)=-1
eq0 \), Khloe is not correct.