Question

1. find the quotient using synthetic division. express the answer as a rational expression. \\(\dfrac{x^3 + 2x^2 - 3x - 4}{x - 2}\\) options: \\(x^2 + 4x + 5 - \dfrac{6}{x - 2}\\), \\(x^2 + 4x + 5 + \dfrac{6}{x - 2}\\), \\(x^2 + 4x + 5 + \dfrac{6}{-2}\\), \\(x^2 + 4x + 5 - \dfrac{6}{2}\\)

Explanation:

Step1: Set up synthetic division

Root of divisor $x-2$ is $2$. Coefficients of dividend: $1, 2, -3, -4$.

2 | 1   2   -3   -4
          2    8   10
    ----------------
      1   4    5    6

Step2: Identify quotient and remainder

Quotient polynomial: $x^2+4x+5$, remainder: $6$.

Step3: Write rational expression

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

Answer:

$\boldsymbol{x^2 + 4x + 5 + \frac{6}{x-2}}$ (matches the third option)