Question

When the function f(x) is divided by x + 1, the quotient is 3x^2 – 9x – 2 and theremainder is 7. Find the function f(x)

Answer 1

Solution:

Given:

$\frac{f(x)}{x+1}=3x^2-9x-2+\frac{7}{x+1}$

To get the function f(x),

We can use the comparison of converting mixed numbers to an improper fraction as shown below;

$a\frac{b}{x}=\frac{(a\times x)+b}{x}$

Hence, following this conversion above,

$\begin{gathered} \frac{f(x)}{x+1}=\frac{(3x^2-9x-2)(x+1)+7}{x+1} \\ \frac{f(x)}{x+1}=\frac{(3x^3-9x^2-2x+3x^2-9x-2)+7}{x+1} \\ \frac{f(x)}{x+1}=\frac{(3x^3-9x^2+3x^2-2x-9x-2)+7}{x+1} \\ \frac{f(x)}{x+1}=\frac{(3x^3-6x^2-11x-2)+7}{x+1} \\ \frac{f(x)}{x+1}=\frac{3x^3-6x^2-11x-2+7}{x+1} \\ \frac{f(x)}{x+1}=\frac{3x^3-6x^2-11x+5}{x+1} \\ \\ \text{Comparing both sides of the equation, then } \\ f(x)=3x^3-6x^2-11x+5 \end{gathered}$

Therefore, the function f(x) is;

$3x^3-6x^2-11x+5$