Question

evaluate each function at the given value. using the remainder theorem

1. $f(x) = -3x^3 + 9x^2 - 3x - 1$ at $x = 2$

Explanation:

Step1: Recall Remainder Theorem

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). So to find \( f(2) \) for \( f(x)=-3x^{3}+9x^{2}-3x - 1 \), we substitute \( x = 2 \) into the polynomial.

Step2: Substitute \( x = 2 \) into \( f(x) \)

Calculate each term:

• For the \( -3x^{3} \) term: \( -3\times(2)^{3}=-3\times8=-24 \)
• For the \( 9x^{2} \) term: \( 9\times(2)^{2}=9\times4 = 36 \)
• For the \( -3x \) term: \( -3\times2=-6 \)
• The constant term: \( -1 \)

Step3: Sum the terms

Add all the calculated terms together: \( -24 + 36-6 - 1 \)
First, \( -24+36 = 12 \), then \( 12-6 = 6 \), then \( 6 - 1=5 \)

Answer:

\( 5 \)