Questions: Use the remainder theorem to find the remainder when f(x) is divided by x-2. Then use f(x) = 3x^3 - 9x^2 + 15x - 12 The remainder is .

Transcript text: Use the remainder theorem to find the remainder when $f(x)$ is divided by $x-2$. Then use \[ f(x)=3 x^{3}-9 x^{2}+15 x-12 \] The remainder is $\square$.

Solution

Solution Steps

To find the remainder when $ f(x) $ is divided by $ x-2 $ using the Remainder Theorem, we need to evaluate $ f(2) $. The Remainder Theorem states that the remainder of the division of a polynomial $ f(x) $ by $ x - c $ is $ f(c) $.

Step 1: Define the Polynomial

We are given the polynomial function: \[ f(x) = 3x^3 - 9x^2 + 15x - 12 \]

Step 2: Apply the Remainder Theorem

To find the remainder when $ f(x) $ is divided by $ x - 2 $, we evaluate $ f(2) $: \[ f(2) = 3(2)^3 - 9(2)^2 + 15(2) - 12 \]

Step 3: Calculate $ f(2) $

Calculating each term: \[ f(2) = 3(8) - 9(4) + 30 - 12 \] \[ = 24 - 36 + 30 - 12 \] \[ = 24 - 36 + 30 - 12 = 6 \]