Question

use the remainder theorem to determine if $x - 2$ is a factor of the polynomial $f(x)=3x^{5}-7x^{3}-11x^{2}+2$.

Explanation:

Step1: Recall the Remainder Theorem

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - c \), then the remainder is \( f(c) \). Also, \( x - c \) is a factor of \( f(x) \) if and only if \( f(c)=0 \). Here, \( c = 2 \) (since we are checking \( x - 2 \) as a factor), so we need to compute \( f(2) \).

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

Given \( f(x)=3x^{5}-7x^{3}-11x^{2}+2 \), substitute \( x = 2 \):
\[

$$\begin{align*} f(2)&=3(2)^{5}-7(2)^{3}-11(2)^{2}+2\\ &=3(32)-7(8)-11(4)+2\\ &=96 - 56 - 44 + 2 \end{align*}$$

\]

Step3: Simplify the expression

First, calculate \( 96 - 56 = 40 \). Then, \( 40 - 44=-4 \). Finally, \( -4 + 2=-2 \).

Answer:

Since \( f(2)=-2
eq0 \), by the Remainder Theorem, \( x - 2 \) is not a factor of the polynomial \( f(x)=3x^{5}-7x^{3}-11x^{2}+2 \).