If f, of, x, equals, 5, x, to the power 5 , plus, 3, x, cubed, plus, 2f(x)=5x

Question

5
+3x
3
+2, 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

To find the remainder when \( f(x) = 5x^5 + 3x^3 + 2 \) is divided by \( x - 2 \), we will use the Remainder Theorem which states that the remainder of \( f(x) \) divided by \( x - c \) is \( f(c) \).

1. We need to evaluate \( f(2) \): \[ f(2) = 5(2^5) + 3(2^3) + 2 \]

   Now calculating each term: \[ 2^5 = 32 \quad \Rightarrow \quad 5(32) = 160 \] \[ 2^3 = 8 \quad \Rightarrow \quad 3(8) = 24 \] \[ \text{So, } f(2) = 160 + 24 + 2 = 186 \]

2. Therefore, the remainder when \( f(x) \) is divided by \( x - 2 \) is \( 186 \).

Next, we are tasked with finding the remainder when \( f(x) \) is divided by \( x - 2 \) using the above steps.

1. According to the remainder theorem, the remainder of \( f(x) \) divided by \( (x-2) \) is equal to: \[ f(2) = 186 \]

In summary, the remainder when \( f(x) = 5x^5 + 3x^3 + 2 \) is divided by \( x - 2 \) is \( \boxed{186} \).