Question

use synthetic division to find the quotient.
$\frac{3x^4 - 5x^3 + 7}{x + 1}$
$?x^3 + x^2 + x + + \frac{}{x+1}$

Explanation:

Step1: Identify root & coefficients

Divisor root: $x = -1$; Dividend coefficients: $3, -5, 0, 0, 7$

Step2: Bring down leading coefficient

$$\begin{array}{r|rrrrr} -1 & 3 & -5 & 0 & 0 & 7 \\ \hline & & & & & \\ \hline & 3 & & & & \end{array}$$

Step3: Multiply & add (1st term)

Multiply $3 \times -1 = -3$, add to $-5$: $-5 + (-3) = -8$

$$\begin{array}{r|rrrrr} -1 & 3 & -5 & 0 & 0 & 7 \\ \hline & & -3 & & & \\ \hline & 3 & -8 & & & \end{array}$$

Step4: Multiply & add (2nd term)

Multiply $-8 \times -1 = 8$, add to $0$: $0 + 8 = 8$

$$\begin{array}{r|rrrrr} -1 & 3 & -5 & 0 & 0 & 7 \\ \hline & & -3 & 8 & & \\ \hline & 3 & -8 & 8 & & \end{array}$$

Step5: Multiply & add (3rd term)

Multiply $8 \times -1 = -8$, add to $0$: $0 + (-8) = -8$

$$\begin{array}{r|rrrrr} -1 & 3 & -5 & 0 & 0 & 7 \\ \hline & & -3 & 8 & -8 & \\ \hline & 3 & -8 & 8 & -8 & \end{array}$$

Step6: Multiply & add (remainder)

Multiply $-8 \times -1 = 8$, add to $7$: $7 + 8 = 15$

$$\begin{array}{r|rrrrr} -1 & 3 & -5 & 0 & 0 & 7 \\ \hline & & -3 & 8 & -8 & 8 \\ \hline & 3 & -8 & 8 & -8 & 15 \end{array}$$

Answer:

$3x^3 -8x^2 +8x -8 + \frac{15}{x+1}$