Question

When the function f, of, xf(x) is divided by x, minus, 2x−2, the quotient is 2, x, squared, plus, 8, x, plus, 52x
2
+8x+5 and the remainder is 22. Find the function f, of, xf(x) and write the result in standard form.


f, of, x, equalsf(x)=











Solve the problem just like the example below
⬇️
When the function
f, of, x
f(x) is divided by
x, minus, 2
x−2, the quotient is
x, squared, plus, x, plus, 1
x
2
+x+1 and the remainder is
minus, 5
−5. Find the function
f, of, x
f(x) and write the result in standard form.
start fraction, dividend, divided by, divisor, end fraction, equals, quotient, plus, start fraction, remainder, divided by, divisor, end fraction
divisor
dividend
​
=quotient+
divisor
remainder
​
start fraction, f, of, x, divided by, x, minus, 2, end fraction, equals, left parenthesis, x, squared, plus, x, plus, 1, right parenthesis, plus, start fraction, minus, 5, divided by, x, minus, 2, end fraction
x−2
f(x)
​
=(x
2
+x+1)+
x−2
−5
​
left parenthesis, x, minus, 2, right parenthesis, left parenthesis, start fraction, f, of, x, divided by, x, minus, 2, end fraction, right parenthesis, equals, left parenthesis, left parenthesis, x, squared, plus, x, plus, 1, right parenthesis, plus, start fraction, minus, 5, divided by, x, minus, 2, end fraction, right parenthesis, left parenthesis, x, minus, 2, right parenthesis
(x−2)(
x−2
f(x)
​
)=((x
2
+x+1)+
x−2
−5
​
)(x−2)
f, of, x, equals, left parenthesis, x, minus, 2, right parenthesis, left parenthesis, x, squared, plus, x, plus, 1, right parenthesis, minus, 5
f(x)=(x−2)(x
2
+x+1)−5

x, squared
x
2
plus, x
+x
plus, 1
+1
x
x
x, cubed
x
3
x, squared
x
2
x
x
minus, 2
−2
minus, 2, x, squared
−2x
2
minus, 2, x
−2x
minus, 2
−2
f, of, x, equals, left parenthesis, x, cubed, minus, x, squared, minus, x, minus, 2, right parenthesis, minus, 5
f(x)=(x
3
−x
2
−x−2)−5
f, of, x, equals, x, cubed, minus, x, squared, minus, x, minus, 7
f(x)=x
3
−x
2
−x−7

Answer 1

Let's solve the problem step-by-step based on the given structure.

Given:
- The divisor is \( x - 2 \).
- The quotient is \( 2x^2 + 8x + 5 \).
- The remainder is \( 22 \).

### 1. Write the division equation:

The general form of polynomial long division can be expressed as:

\[
\frac{f(x)}{x - 2} = \text{quotient} + \frac{\text{remainder}}{\text{divisor}}
\]

So we have:

\[
\frac{f(x)}{x - 2} = (2x^2 + 8x + 5) + \frac{22}{x - 2}
\]

### 2. Multiply both sides by \( x - 2 \) to get \( f(x) \):

\[
f(x) = (2x^2 + 8x + 5)(x - 2) + 22
\]

### 3. Distribute \( (2x^2 + 8x + 5)(x - 2) \):

First, distribute:

\[
f(x) = (2x^2 + 8x + 5)(x) - (2x^2 + 8x + 5)(2)
\]

Calculating the first term:

\[
(2x^2 + 8x + 5)(x) = 2x^3 + 8x^2 + 5x
\]

Calculating the second term:

\[
(2x^2 + 8x + 5)(2) = 4x^2 + 16x + 10
\]

Now, putting this together:

\[
f(x) = (2x^3 + 8x^2 + 5x) - (4x^2 + 16x + 10) + 22
\]

### 4. Combine like terms:

Combine the like terms:

1. **Cubic term**: \( 2x^3 \)
2. **Quadratic terms**: \( 8x^2 - 4x^2 = 4x^2 \)
3. **Linear terms**: \( 5x - 16x = -11x \)
4. **Constant terms**: \( 22 - 10 = 12 \)

Putting this all together, we get:

\[
f(x) = 2x^3 + 4x^2 - 11x + 12
\]

### Final Answer

Therefore, the function \( f(x) \) in standard form is:

\[
\boxed{2x^3 + 4x^2 - 11x + 12}
\]