When the function f, of, xf(x) is divided by x, plus, 5x+5, the quotient is 3, x, squared, minus, 4, x, plus, 23x

Question

2
−4x+2 and the remainder is 77. Find the function f, of, xf(x) and write the result in standard form.
Answer
Attempt 1 out of 5

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 solve the problem, we need to follow the given example's structure. We know that \( f(x) \) when divided by \( x + 5 \) gives a quotient of \( 3x^2 - 4x + 2 \) and a remainder of \( 77 \).

Using the polynomial long division structure, we represent the division as follows:

\[ \frac{f(x)}{x + 5} = 3x^2 - 4x + 2 + \frac{77}{x + 5} \]

Now, rearranging gives:

\[ f(x) = (x + 5) \left(3x^2 - 4x + 2\right) + 77 \]

Next, we need to perform the multiplication:

1. Distribute \( (x + 5) \) with \( (3x^2 - 4x + 2) \): \[ (x + 5)(3x^2 - 4x + 2) = x(3x^2 - 4x + 2) + 5(3x^2 - 4x + 2) \]

   • For \( x(3x^2 - 4x + 2) \): \[ = 3x^3 - 4x^2 + 2x \]

   • For \( 5(3x^2 - 4x + 2) \): \[ = 15x^2 - 20x + 10 \]

   Now combine the results: \[ 3x^3 + (-4x^2 + 15x^2) + (2x - 20x) + 10 = 3x^3 + 11x^2 - 18x + 10 \]

2. Now, add the remainder \( 77 \): \[ f(x) = 3x^3 + 11x^2 - 18x + 10 + 77 \] \[ f(x) = 3x^3 + 11x^2 - 18x + 87 \]

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

\[ \boxed{3x^3 + 11x^2 - 18x + 87} \]