Question

What is the remainder of 3 2 ( 6 9 3) ( 3) xxx x − − +÷− ?
A. –51
B. 51
x 3
−
+
C. 17
x
−
D. 17
x 3

Answer 1

To solve for the remainder of the expression \( \frac{3x^2 - (6x^3 + 3)}{x - 3} \), we can apply the Remainder Theorem. According to this theorem, the remainder of a polynomial \( P(x) \) when divided by \( x - c \) is \( P(c) \).

Here, \( P(x) = 3x^2 - (6x^3 + 3) \) which simplifies to \( P(x) = -6x^3 + 3x^2 - 3 \).

We want to find the remainder when \( P(x) \) is divided by \( x - 3 \). Therefore, we need to evaluate \( P(3) \):

\[
P(3) = -6(3)^3 + 3(3)^2 - 3
\]

Calculating each term:

1. \( -6(3)^3 = -6(27) = -162 \)
2. \( 3(3)^2 = 3(9) = 27 \)
3. The last term is simply \( -3 \).

Now, we combine these results:

\[
P(3) = -162 + 27 - 3
\]
\[
P(3) = -162 + 24
\]
\[
P(3) = -138
\]

Thus, the remainder when \( 3x^2 - (6x^3 + 3) \) is divided by \( x - 3 \) is:

The answer is \( \boxed{-138} \).

None of the provided answer choices match, so it seems there might have been an error in the question or selection of answer choices.