Question

1. perform the following operation (f · g)(x)

$f(x) = 2x^2 - 3x + 5$
$g(x) = 3x - 14$

Explanation:

Step1: Understand the operation

The operation \((f \cdot g)(x)\) means the product of the functions \(f(x)\) and \(g(x)\), so we need to multiply \(f(x) = 2x^2 - 3x + 5\) and \(g(x) = 3x - 14\).

Step2: Multiply the two polynomials

We use the distributive property (FOIL method extended for polynomials):
\[

$$\begin{align*} (f \cdot g)(x)&=(2x^2 - 3x + 5)(3x - 14)\\ &=2x^2(3x) + 2x^2(-14) - 3x(3x) - 3x(-14) + 5(3x) + 5(-14)\\ &=6x^3 - 28x^2 - 9x^2 + 42x + 15x - 70 \end{align*}$$

\]

Step3: Combine like terms

Combine the \(x^2\) terms and the \(x\) terms:
\[

$$\begin{align*} &6x^3 + (-28x^2 - 9x^2) + (42x + 15x) - 70\\ &=6x^3 - 37x^2 + 57x - 70 \end{align*}$$

\]

Answer:

\(6x^3 - 37x^2 + 57x - 70\)