Question

For functions $f\left(x\right)=x^3+x^2-7x+2$and $g\left(x\right)=x-2$.

Find (a)$\left(\frac{f}{g}\right)\left(x\right)$and (b)$\left(\frac{f}{g}\right)\left(2\right)$.

Short answer

Part (a). The solution is $x^2+3x-1$.

Part (b). The solution is$9$.

Step-by-step solution

Part (a). Step 1. Given information.

The functions$f\left(x\right)=x^3+x^2-7x+2$and$g\left(x\right)=x-2$.

Part (a). Step 2. To divide the polynomial function by substituting the fx and gx in fgx.

Divide the given polynomials.

$$\begin{gathered} x-2\begin{array}{ccc}{x}^2& +3x& -1\end{array}\begin{array}{cccc}{x}^3& +x^2& -7x& +2\\ x^3& -2x^2& & \end{array} \\ \overline{3x^2-7x} \\ 3x^2-6x \\ \overline{-x+2} \\ -x+2 \\ \overline{0} \end{gathered}$$

$$\begin{gathered} x-2\begin{array}{ccc}{x}^2& +3x& -1\end{array}\begin{array}{cccc}{x}^3& +x^2& -7x& +2\\ x^3& -2x^2& & \end{array} \\ \overline{3x^2-7x} \\ 3x^2-6x \\ \overline{-x+2} \\ -x+2 \\ \overline{0} \end{gathered}$$

Part (a). Step 3. Simplified answer.

Hence, the obtained result is$\left(\frac{f}{g}\right)\left(x\right)=x^2+3x-1$.

Part (b). Step 1. Given information.

The functions$f\left(x\right)=x^3+x^2-7x+2$and$g\left(x\right)=x-2$.

Part (b). Step 2. To find the fg2 by substituting the x=2 in fgx=x2+3x-1.

Solve it as follows:

$$\begin{gathered} \left(\frac{f}{g}\right)\left(2\right)={\left(2\right)}^2+3\left(2\right)-1 \\ \Rightarrow 4+6-1 \\ \Rightarrow 9 \end{gathered}$$

Part (b). Step 3. Simplified answer.

Hence, the simplified expression is$9$.