Question

use synthetic division to divide.
\\(\frac{4x^2 - 3}{x - 2}\\)
\\(\frac{4x^2 - 3}{x - 2} = \square\\)

Explanation:

Step1: Identify root & coefficients

Root of divisor $x-2$ is $2$. Coefficients of dividend $4x^2+0x-3$ are $4, 0, -3$.

Step2: Set up synthetic division

Write root on left, coefficients on top:

$$\begin{array}{r|rrr} 2 & 4 & 0 & -3 \\ \hline & & 8 & 16 \\ \hline & 4 & 8 & 13 \end{array}$$

Step3: Interpret results

First two values are quotient coefficients, last is remainder. Quotient: $4x+8$, remainder: $13$.

Step4: Write final expression

$\frac{\text{Dividend}}{\text{Divisor}} = \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}$

Answer:

$4x + 8 + \frac{13}{x-2}$