Question

What is the remainder when
x
3
+
3
x
2
−
10
x
−
14
x
3
+3x
2
−10x−14 is divided by
x
−
3
x−3?

Answer 1

To find the remainder when the polynomial \( x^3 + 3x^2 - 10x - 14 \) is divided by \( x - 3 \), we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the polynomial \( f(x) \) when divided by \( x - c \) is \( f(c) \).

In this case, our polynomial is

\[
f(x) = x^3 + 3x^2 - 10x - 14
\]

We need to find \( f(3) \):

\[
f(3) = 3^3 + 3(3^2) - 10(3) - 14
\]

Calculating each term:

\[
3^3 = 27
\]
\[
3(3^2) = 3(9) = 27
\]
\[
-10(3) = -30
\]

Now substituting these values into \( f(3) \):

\[
f(3) = 27 + 27 - 30 - 14
\]

Now, perform the additions and subtractions:

\[
27 + 27 = 54
\]
\[
54 - 30 = 24
\]
\[
24 - 14 = 10
\]

Thus, the remainder when \( x^3 + 3x^2 - 10x - 14 \) is divided by \( x - 3 \) is

\[
\boxed{10}
\]