Question

When f(x)=x^4−5x^3+8x is divided by x−3 , its remainder is −30 . Which of the following is correct according to the Remainder Theorem? (1 point)
f(-3)=-30
f(3)=-30
f(3)=30
f(-3)=30

Answer 1

According to the Remainder Theorem, when a polynomial \( f(x) \) is divided by \( x - c \), the remainder of that division is equal to \( f(c) \).

In this case, we are given that when \( f(x) = x^4 - 5x^3 + 8x \) is divided by \( x - 3 \), the remainder is \( -30 \). This means:

\[
f(3) = -30
\]

Now, let's analyze the options given:

1. \( f(-3) = -30 \) — This is not supported by the information given.
2. \( f(3) = -30 \) — This is correct based on the Remainder Theorem.
3. \( f(3) = 30 \) — This is incorrect because we established from the theorem that \( f(3) = -30 \).
4. \( f(-3) = 30 \) — This is also not supported by the information given.

Thus, the correct choice according to the Remainder Theorem is:

\[
\boxed{f(3) = -30}
\]