Question

Use the Remainder Theorem to determine if x minus 2 is a factor of the polynomial f left parenthesis x right parenthesis equals 3 x superscript 5 baseline minus 7 x cubed minus 11 x squared plus 2.

Answer 1

To determine if \( x - 2 \) is a factor of the polynomial \( f(x) = 3x^5 - 7x^3 - 11x^2 + 2 \) using the Remainder Theorem, we need to evaluate the polynomial at \( x = 2 \). According to the Remainder Theorem, if \( f(2) = 0 \), then \( x - 2 \) is a factor of \( f(x) \).

Now, let's calculate \( f(2) \):

\[
f(2) = 3(2^5) - 7(2^3) - 11(2^2) + 2
\]

Calculating each term step-by-step:

1. \( 2^5 = 32 \), so \( 3(2^5) = 3 \times 32 = 96 \)
2. \( 2^3 = 8 \), so \( -7(2^3) = -7 \times 8 = -56 \)
3. \( 2^2 = 4 \), so \( -11(2^2) = -11 \times 4 = -44 \)

Now we can combine these results:

\[
f(2) = 96 - 56 - 44 + 2
\]

Calculating this step-by-step:

- Start with \( 96 - 56 = 40 \)
- Then \( 40 - 44 = -4 \)
- Finally \( -4 + 2 = -2 \)

Thus, we find that:

\[
f(2) = -2
\]

Since \( f(2) \neq 0 \), it follows that \( x - 2 \) is **not** a factor of the polynomial \( f(x) \).