Question

given $f(x) = 3x^3 + kx - 13$, and the remainder when $f(x)$ is divided by $x - 3$ is 95, 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 \( f(x) \) by \( x - 3 \), so \( a = 3 \), and the remainder is \( f(3) = 95 \).

Step2: Substitute \( x = 3 \) into \( f(x) \)

Given \( f(x) = 3x^3 + kx - 13 \), substitute \( x = 3 \):

$$ f(3) = 3(3)^3 + k(3) - 13 $$

Calculate \( 3^3 = 27 \), so:

$$ f(3) = 3(27) + 3k - 13 $$

$$ f(3) = 81 + 3k - 13 $$

Simplify \( 81 - 13 = 68 \):

$$ f(3) = 68 + 3k $$

Step3: Solve for \( k \)

We know \( f(3) = 95 \), so set up the equation:

$$ 68 + 3k = 95 $$

Subtract 68 from both sides:

$$ 3k = 95 - 68 $$

$$ 3k = 27 $$

Divide both sides by 3:

$$ k = \frac{27}{3} $$

$$ k = 9 $$

Answer:

\( k = 9 \)