Question

use the remainder theorem and synthetic division to find f(k).
k = -2; f(x) = 4x3 - 5x2 - 3x + 24
a -22
b -34
c -42
d -52

Explanation:

Step1: Recall Remainder Theorem

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - k \), the remainder is \( f(k) \). We can use synthetic division to find the remainder when \( f(x) = 4x^3 - 5x^2 - 3x + 24 \) is divided by \( x - (-2)=x + 2 \).

Step2: Set up Synthetic Division

For synthetic division with \( k=-2 \), we use the coefficients of \( f(x) \): \( 4 \) (coefficient of \( x^3 \)), \( -5 \) (coefficient of \( x^2 \)), \( -3 \) (coefficient of \( x \)), and \( 24 \) (constant term).
Set up the synthetic division as follows:

$$
LATEXBLOCK0
$$

• Bring down the leading coefficient \( 4 \).
• Multiply \( 4 \) by \( -2 \) to get \( -8 \), add to \( -5 \): \( -5 + (-8)=-13 \).
• Multiply \( -13 \) by \( -2 \) to get \( 26 \), add to \( -3 \): \( -3 + 26 = 23 \).
• Multiply \( 23 \) by \( -2 \) to get \( -46 \), add to \( 24 \): \( 24 + (-46)=-22 \).

The remainder is \( -22 \), so by the Remainder Theorem, \( f(-2)=-22 \).

Answer:

A. -22