Q.328. - Intermediate Algebra

Question

For functions $f\left(x\right)=x^2-5x+2$ and $g\left(x\right)=x^2-3x-1$.

Find (a) $\left(f·g\right)\left(x\right)$ and (b) $\left(f·g\right)\left(-1\right)$.

Step-by-Step Solution

Answer

Part (a). The solution is $x^4-8x^3+16x^2-x-2$.

Part (b). The solution is $24$.

Part (a). Step 1. Given information.

The functions $f\left(x\right)=x^2-5x+2$ and $g\left(x\right)=x^2-3x-1$.

Part (a). Step 2. Multiply the polynomial functions.

$$\begin{gathered} \left(f·g\right)\left(x\right)=f\left(x\right)·g\left(x\right) \\ \Rightarrow \left(x^2-5x+2\right)·\left(x^2-3x-1\right) \\ \Rightarrow x^2\left(x^2-3x-1\right)-5x\left(x^2-3x-1\right)+2\left(x^2-3x-1\right) \\ \Rightarrow x^4-3x^3-x^2-5x^3+15x^2+5x+2x^2-6x-2 \\ \Rightarrow x^4-8x^3+16x^2-x-2 \end{gathered}$$

Part (a). Step 3. Simplified answer.

Hence, the obtained result is $x^4-8x^3+16x^2-x-2$.$x^4-8x^3+16x^2-x-2$

Part (b). Given information.

The functions $f\left(x\right)=\left(x^2-5x+2\right)$ and $g\left(x\right)=\left(x^2-3x-1\right)$.

Part (b). Step 2. To find the value of f · g - 1 by substituting x = - 1 in f · g x = x 4 - 8 x 3 + 16 x 2 - x - 2

Solve it as follows:

$$\begin{gathered} \left(f·g\right)\left(-1\right)={\left(-1\right)}^4-8{\left(-1\right)}^3+16{\left(-1\right)}^2-\left(-1\right)-2 \\ \Rightarrow 1-8\left(-1\right)+16+1-2 \\ \Rightarrow 1+8+16-1 \\ \Rightarrow 24 \end{gathered}$$

Part (b). Simplified answer.

Hence, the simplified expression is $24$.