Q.326. - Intermediate Algebra

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) $\left(\frac{f}{g}\right)\left(2\right)$.

Step-by-Step Solution

Answer

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

Part (b). The solution is $9$.

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 f x and g x in f g x .

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 f g 2 by substituting the x = 2 in f g x = x 2 + 3 x - 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 $9$.