If f, of, x, equals, 3, x, to the power 4 , plus, 4f(x)=3x

Question

4
+4, then what is the remainder when f, of, xf(x) is divided by x, minus, 2x−2?

Answer 1

To solve for the remainder when \( f(x) \) is divided by \( x - 2 \), we first need to express \( f(x) \) given the equation:

\[ f(x) = 3x^4 + 4. \]

We need to find \( f(x) \) first:

1. Calculate \( f(2) \): \[ f(2) = 3(2^4) + 4 = 3(16) + 4 = 48 + 4 = 52. \]

2. Now we need to find \( f(x) - 52 \) and express it in terms of \( x - 2 \): \[ f(x) - 52 = 3x^4 + 4 - 52 = 3x^4 - 48. \]

3. We can factor \( 3x^4 - 48 \): \[ 3(x^4 - 16) = 3(x^2 - 4)(x^2 + 4) = 3(x - 2)(x + 2)(x^2 + 4). \]

4. Therefore, when dividing \( f(x) \) by \( x - 2 \), we find that the remainder is: \[ r = 52. \]

So the final answer is:

\[ \text{The remainder when } f(x) \text{ is divided by } x - 2 \text{ is } 52. \]