Question

(a) complete this synthetic division table.
$\

$$\begin{array}{r} 3\\enclose{longdiv}{-3\\ \\ 0\\ \\ 19\\ \\ 24} \\\\\\ \\ \\ \\ \\ \\boxed{\\phantom{0}}\\ \\boxed{\\phantom{0}}\\ \\boxed{\\phantom{0}} \\\\\\ \\hline \\boxed{\\phantom{0}}\\ \\boxed{\\phantom{0}}\\ \\boxed{\\phantom{0}}\\ \\boxed{\\phantom{0}} \\end{array}$$

$
(b) write your answer in the following form: $\text{quotient}+\frac{\text{remainder}}{x-3}$.
$\frac{-3x^{3}+19x + 24}{x - 3}=\boxed{\phantom{0}}+\frac{\boxed{\phantom{0}}}{x - 3}$

Explanation:

Step1: Bring down leading coefficient

Bring down $-3$ to the result row.

Step2: Multiply by divisor, add next term

Multiply $-3$ by $3$: $-3 \times 3 = -9$. Add to next coefficient: $0 + (-9) = -9$.

Step3: Multiply by divisor, add next term

Multiply $-9$ by $3$: $-9 \times 3 = -27$. Add to next coefficient: $19 + (-27) = -8$.

Step4: Multiply by divisor, add last term

Multiply $-8$ by $3$: $-8 \times 3 = -24$. Add to last coefficient: $24 + (-24) = 0$.

Filled synthetic division table:

3 ) -3   0   19   24
        -9  -27  -24
      ----------------
      -3  -9   -8    0

Step5: Identify quotient and remainder

The quotient is the polynomial with coefficients from the result row (excluding the last term), degree reduced by 1: $-3x^2 -9x -8$. The remainder is the last term: $0$.

Answer:

(a)

3 ) -3   0   19   24
        -9  -27  -24
      ----------------
      -3  -9   -8    0

(b)
$\frac{-3x^3 + 19x + 24}{x-3} = -3x^2 -9x -8 + \frac{0}{x-3}$