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

Question

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

To find the function \( f(x) \) given that when \( f(x) \) is divided by \( x - 2 \), the quotient is \( 2x^2 + 8x + 5 \) and the remainder is \( 22 \), we will follow a similar method to the provided example.

1. Start with the division algorithm: We can express the relationship between the dividend, divisor, quotient, and remainder as follows:

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

2. Multiply both sides by the divisor \( x - 2 \):

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

3. Now, distribute \( (2x^2 + 8x + 5)(x - 2) \):

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

4. Calculating \( (2x^2 + 8x + 5)x \):

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

5. Calculating \( (2x^2 + 8x + 5)2 \):

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

6. Subtract \( 4x^2 + 16x + 10 \) from \( 2x^3 + 8x^2 + 5x \) and add the remainder \( 22 \):

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

7. Combine like terms:

The full expression is:

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

This simplifies to:

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

8. Final result:

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

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