Question

1. find the quotient using synthetic division. express the answer as a rational expression. \\(dfrac{3x^4 + 12x^3 - 5x^2 + 40x - 40}{x + 5}\\) options: \\(3x^3 - 3x^2 + 10x - 10 - dfrac{10}{5}\\), \\(3x^3 - 3x^2 + 10x - 10 + dfrac{10}{-5}\\), \\(3x^3 - 3x^2 + 10x - 10 + dfrac{10}{x + 5}\\), \\(3x^3 - 3x^2 + 10x - 10 - dfrac{10}{x + 5}\\)

Explanation:

Step1: Identify root and coefficients

Divisor root: $x=-5$; Dividend coefficients: $3, 12, -5, 40, -40$

Step2: Set up synthetic division

$$\begin{array}{r|rrrrr} -5 & 3 & 12 & -5 & 40 & -40 \\ & & -15 & 15 & -50 & 50 \\ \hline & 3 & -3 & 10 & -10 & 10 \\ \end{array}$$

Step3: Form quotient and remainder

Quotient polynomial: $3x^3 - 3x^2 + 10x - 10$; Remainder: $10$

Step4: Write rational expression

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

Answer:

$\boldsymbol{3x^3 - 3x^2 + 10x - 10 + \frac{10}{x+5}}$ (matches the second option)