Finding Remainder of Polynomial Division Using the Remainder Theorem
The question asks for the remainder when the polynomial \(-x^3 + x^2 + 5x - 6\) is divided by the binomial \(x + 3\).
To find the remainder without performing the entire polynomial long division, we can use the Remainder Theorem. According to the Remainder Theorem, if a polynomial \(f(x)\) is divided by \(x - a\), the remainder is \(f(a)\).
Since we're dividing by \(x + 3\), we can apply the Remainder Theorem by substituting \(x = -3\) into the polynomial:
\[
f(-3) = -(-3)^3 + (-3)^2 + 5(-3) - 6
\]
\[
f(-3) = -(-27) + 9 - 15 - 6
\]
\[
f(-3) = 27 + 9 - 15 - 6
\]
\[
f(-3) = 36 - 21
\]
\[
f(-3) = 15
\]
So the remainder is 15, which corresponds to option B.
