Question

If f, of, x, equals, x, to the power 6 , plus, x, minus, 1f(x)=x
6
+x−1, then what is the remainder when f, of, xf(x) is divided by x, plus, 3x+3?





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

To solve for the remainder when \( f(x) = x^6 + x - 1 \) is evaluated at \( f(x) \) and divided by \( x + 3 \), we will use the Remainder Theorem.

**Step 1: Identify the polynomial and the divisor.**
Here, we want to find the remainder when \( f(f(x)) \) is divided by \( x + 3 \).

**Step 2: Use the Remainder Theorem.**
According to the Remainder Theorem, the remainder of \( f(x) \) divided by \( x - c \) is \( f(c) \). Thus, we need to evaluate \( f(f(-3)) \).

**Step 3: Calculate \( f(-3) \).**
First, compute \( f(-3) \):
\[
f(-3) = (-3)^6 + (-3) - 1
\]
Calculating step-by-step:
- \( (-3)^6 = 729 \)
- \( -3 - 1 = -4 \)

So,
\[
f(-3) = 729 - 4 = 725
\]

**Step 4: Evaluate \( f(f(-3)) = f(725) \).**
Next, we find \( f(725) \):
\[
f(725) = 725^6 + 725 - 1
\]

**Step 5: Now we find the remainder of \( f(725) \) when divided by \( x + 3 \).**
Instead of calculating \( f(725) \) entirely, we must find the form it takes when divided by \( x + 3 \). By the Remainder Theorem, calculate \( f(-3) \) again but with the new value.
\[
f(x) \mod (x + 3) \ (c = -3)
\]
Using the general form of \( f(x) \):
\[
= x^6 + x - 1
\]
When evaluated at \( x = -3 \):
\[
f(-3) = 725
\]

Since \( f(x) \) is not a simple one to compute directly, instead we could find \( f(725) \mod (x + 3) \) directly by recognizing the quartic powers will cycle through mod:
Let us compute directly:
- Given \( x + 3 = 0 \implies x = -3 \), we substitute this in:
Thus find \( f(-3) = 725 \) again gives 725.

But as stated follow the polynomial division rules we can only find final terms when cycling:

The final answer is instead tracking it again:
### Final Result:
The remainder when \( f(f(x)) \) is divided by \( x + 3 \) is **\(\equiv 725 \)**.