Question

given $f(x) = 2x^3 - 5x + k$, and the remainder when $f(x)$ is divided by $x - 1$ is 14, then what is the value of $k$?

Explanation:

Step1: Recall the Remainder Theorem

The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). Here, we are dividing \( f(x) = 2x^3 - 5x + k \) by \( x - 1 \), so \( a = 1 \), and the remainder is \( f(1) \).

Step2: Substitute \( x = 1 \) into \( f(x) \)

Substitute \( x = 1 \) into \( f(x) \): \( f(1)=2(1)^3 - 5(1)+k \).
Simplify the expression: \( f(1)=2 - 5 + k \), which simplifies to \( f(1)=k - 3 \).

Step3: Set the remainder equal to 14 and solve for \( k \)

We know the remainder is 14, so \( k - 3 = 14 \).
Add 3 to both sides of the equation: \( k=14 + 3 \).

Answer:

\( k = 17 \)