Q.327. - Intermediate Algebra

Question

For functions $f\left(x\right)=x^3+2x^2-19x+12$ and $g\left(x\right)=x-3$.

Find (a) $\left(\frac{f}{g}\right)\left(x\right)$ and (b) $\left(\frac{f}{g}\right)\left(0\right)$.

Step-by-Step Solution

Answer

Part (a). The solution is $x^2+5x-4$.

Part (b). The solution is $-4$.

Part (a). Step 1. Given information.

The functions $f\left(x\right)=x^3+2x^2-19x+12$ and $g\left(x\right)=x-3$.

Part (b). Step 2. To divide the polynomial function by substituting the f x and g x in f g x .

$$\begin{gathered} \left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)} \\ \Rightarrow \frac{x^3+2x^2-19x+12}{x-3} \end{gathered}$$

Part (a). Step 3. Divide the polynomials.

$$\begin{gathered} x-3\begin{array}{ccc}{x}^2& +5x& -4\end{array}\begin{array}{cccc}{x}^3& +2x^2& -19x& +12\\ x^3& -3x^2& & \end{array} \\ \overline{ 5x^2-19x } \\ 5x^2-15x \\ \overline{ -4x+12 } \\ -4x+12 \\ \overline{ 0 } \end{gathered}$$

Part (a). Step (4). Simplified answer.

Hence, the obtained result is $x^2+5x-4$.

Part (b). Step 1. Given information.

The functions $f\left(x\right)=x^3+2x^2-19x+12$ and $g\left(x\right)=x-3$.

Part (b). Step 2. To find the value of f g 0 by substituting the value of x = 0 in f g x = x 2 + 5 x - 4 .

Solve it as follows:

$$\begin{gathered} \left(\frac{f}{g}\right)\left(0\right)=0^2+5\left(0\right)-4 \\ \Rightarrow 0+0-4 \\ \Rightarrow -4 \end{gathered}$$

Part (b). Step 3. Simplified answer.

Hence, the simplified expression is $-4$.