Question

use synthetic division to divide the two polynomials.
\\(w - 2 \overline{)4w^3 - 8w^2 - 3w + 6}\\)
is the divisor given in (x - r) form? yes
how many terms are in the dividend? 4
enter the quotient and remainder.
quotient
+
remainder
synthetic division table
2

Explanation:

Step1: List dividend coefficients

Coefficients of $4w^3 - 8w^2 - 3w + 6$: $4, -8, -3, 6$

Step2: Set up synthetic division

Use $r=2$ (from $w-2$), write:

$$\begin{array}{r|rrrr} 2 & 4 & -8 & -3 & 6 \\ \hline & & 8 & 0 & -6 \\ \hline & 4 & 0 & -3 & 0 \end{array}$$

Step3: Calculate intermediate values

Bring down 4. Multiply $4\times2=8$, add to $-8$: $-8+8=0$.
Multiply $0\times2=0$, add to $-3$: $-3+0=-3$.
Multiply $-3\times2=-6$, add to $6$: $6+(-6)=0$.

Step4: Identify quotient and remainder

Quotient: $4w^2 + 0w - 3 = 4w^2 - 3$
Remainder: $0$

Answer:

Quotient: $\boldsymbol{4w^2 - 3}$
Remainder: $\boldsymbol{0}$

Filled synthetic division table values (top row: $4, -8, -3, 6$; middle row: $8, 0, -6$; bottom row: $4, 0, -3, 0$)