Question

If f, of, x, equals, 2, x, to the power 5 , plus, 5, x, minus, 1f(x)=2x
5
+5x−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

Let's solve the problem step-by-step, using the remainder theorem and following the format of your example.

We are given the function:
\[ f(x) = 2x^5 + 5x - 1 \]

We need to find the remainder when \( f(x) \) is divided by \( x + 3 \).

### Use the Remainder Theorem:
The remainder of \( f(x) \) when divided by \( (x + 3) \) is equal to \( f(-3) \).

### Calculate \( f(-3) \):
Substituting \( -3 \) into the function:
\[
f(-3) = 2(-3)^5 + 5(-3) - 1
\]

Calculating each term:
- First, calculate \( (-3)^5 \):
\[
(-3)^5 = -243
\]
- Then multiply by 2:
\[
2(-243) = -486
\]
- Next, calculate \( 5(-3) \):
\[
5(-3) = -15
\]
- Now combine all the terms:
\[
f(-3) = -486 - 15 - 1
\]
\[
f(-3) = -486 - 15 - 1 = -502
\]

### Conclusion:
Thus, the remainder when \( f(x) \) is divided by \( x + 3 \) is:
\[
\boxed{-502}
\]