Question

q. cam divided (x³ + 3x² - 4x - 2) by factor of (x - c) using synthetic division. his work is shown above. which best describes his mistake?

Explanation:

Step1: Confirm synthetic division setup

The polynomial is $x^3 + 3x^2 - 4x - 2$, so coefficients are $1, 3, -4, -2$. Divisor is $(x-2)$, so root $r=2$. Setup matches:

$$\begin{array}{r|rrrr} 2 & 1 & 3 & -4 & -2 \\ \hline & & & & \\ \hline & & & & \end{array}$$

Step2: Bring down leading coefficient

Bring down the 1:

$$\begin{array}{r|rrrr} 2 & 1 & 3 & -4 & -2 \\ \hline & & & & \\ \hline & 1 & & & \end{array}$$

Step3: Multiply and add to next coefficient

Multiply $1 \times 2 = 2$, add to 3: $3+2=5$:

$$\begin{array}{r|rrrr} 2 & 1 & 3 & -4 & -2 \\ \hline & & 2 & & \\ \hline & 1 & 5 & & \end{array}$$

Step4: Repeat multiply-add step

Multiply $5 \times 2 = 10$, add to $-4$: $-4+10=6$:

$$\begin{array}{r|rrrr} 2 & 1 & 3 & -4 & -2 \\ \hline & & 2 & 10 & \\ \hline & 1 & 5 & 6 & \end{array}$$

Step5: Final multiply-add for remainder

Multiply $6 \times 2 = 12$, add to $-2$: $-2+12=10$:

$$\begin{array}{r|rrrr} 2 & 1 & 3 & -4 & -2 \\ \hline & & 2 & 10 & 12 \\ \hline & 1 & 5 & 6 & 10 \end{array}$$

All steps match Cam's work, so no errors exist.

Answer:

Cam made no mistakes; his synthetic division is correct, and the remainder is 10.