Finding the Remainder of a Polynomial Using the Remainder Theorem
To find the remainder of the polynomial \(-x^3 + x^2 + 5x - 6\) when divided by \(x + 3\), you can use the Remainder Theorem. According to this theorem, if a polynomial \(f(x)\) is divided by a binomial of the form \(x - c\), the remainder is the value of \(f(c)\).
However, we have \(x + 3\) instead of \(x - c\), so we need to find \(f(-3)\), because \(-3\) is the root of the binomial \(x + 3\).
Let's calculate \(f(-3)\):
\[
f(-3) = -(-3)^3 + (-3)^2 + 5(-3) - 6
\]
\[
f(-3) = -(-27) + 9 - 15 - 6
\]
\[
f(-3) = 27 + 9 - 15 - 6
\]
\[
f(-3) = 36 - 21
\]
\[
f(-3) = 15
\]
So, the remainder when \(-x^3 + x^2 + 5x - 6\) is divided by \(x + 3\) is \(15\).
The correct answer is (B) 15.
