Question

If f, of, x, equals, 3, x, to the power 4 , plus, 4f(x)=3x
4
+4, then what is the remainder when f, of, xf(x) is divided by x, minus, 2x−2?







Solve the problem just like the example below
⬇️
If
f, of, x, equals, 5, x, squared, minus, 4
f(x)=5x
2
−4, then what is the remainder when
f, of, x
f(x) is divided by
x, minus, 2
x−2?
Use the remainder theorem:
Use the remainder theorem:
The remainder of , f, of, x, divided by, left parenthesis, x, minus, 2, right parenthesis, equals , f, of, 2
The remainder of f(x)÷(x−2) equals f(2)
f, of, 2, equals, 5, left parenthesis, 2, right parenthesis, squared, minus, 4
f(2)=5(2)
2
−4
16
16
The slow way:
The slow way:
x, minus, 2
x−2
5, x, squared
5x
2

minus, 4
−4

2
2
5
5
0
0
minus, 4
−4


down arrow
↓
10
10
20
20


5
5
10
10
16
16

Answer 1

Let's solve the problem step by step using the same format as the example:

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

We want to find the remainder when \( f(x) \) is divided by \( x - 2 \).

**Use the remainder theorem:**
The remainder of \( f(x) \) divided by \( (x - 2) \) equals \( f(2) \).

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

So, the remainder when \( f(x) \) is divided by \( x - 2 \) is **52**.

Now we want to find the remainder when \( f(x)f(x) \) is divided by \( x - 2 \).

First, we need to calculate \( f(x)f(x) \):
Since \( f(x) = 3x^4 + 4 \), we have:
\[ f(x)f(x) = (3x^4 + 4)(3x^4 + 4) \]

Instead of expanding \( f(x)f(x) \) completely, we can again use the remainder theorem:
The remainder of \( f(x)f(x) \) divided by \( (x - 2) \) equals \( f(2)f(2) \).

We have already calculated \( f(2) \):
\[ f(2) = 52 \]

Now calculate \( f(2)f(2) \):
\[ f(2)f(2) = 52 \times 52 \]
\[ = 2704 \]

Thus, the remainder when \( f(x)f(x) \) is divided by \( x - 2 \) is **2704**.