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: $$2^5=32\Rightarrow 5(32)=160$$ $$2^3=8\Rightarrow 3(8)=24$$ $$\text{So, }f(2)=160+24+2=186$$

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 $\boxed{{\displaystyle 186}}$.