Question

1. choose the best answer. the quotient (\frac{x^3 + 5x - 6x - 49}{x - 3}) is found using synthetic division to be (image of synthetic division table here). which rational expression correctly expresses the quotient and remainder? options: (x^3 + 8x^2 + 18x + 5); (x^2 + 8x + 18 + \frac{5}{3}); (x^2 + 8x + 18 + \frac{5}{x - 3}); (x^2 + 8x + 18 + \frac{3}{5})

Explanation:

Step1: Identify dividend coefficients

Dividend: $x^3 + 5x^2 -6x -49$, coefficients: $1, 5, -6, -49$

Step2: Interpret synthetic division results

From the table, quotient coefficients are $1, 8, 18$, remainder is $5$.
Quotient polynomial: $x^2 +8x +18$

Step3: Write rational expression

Rational form: $\text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}$, divisor is $x-3$.
Expression: $x^2 +8x +18 + \frac{5}{x-3}$

Answer:

$\boldsymbol{x^2 + 8x + 18 + \frac{5}{x - 3}}$ (corresponding to the second option)