Questions: Divide using long division. State the quotient, q(x), and the remainder, r(x). (8 x^3-2 x^2-14 x-28) ÷ (2 x-4) (8 x^3-2 x^2-14 x-28) ÷ (2 x-4) = 4 x^2+6+(-4)/(2 x-4) (Simplify your answers. Do not factor. Use integers or fractions for any numbers in the expressions.)

$Divide using long division. State the quotient, q(x), and the remainder, r(x). (8 x^3-2 x^2-14 x-28) ÷ (2 x-4) (8 x^3-2 x^2-14 x-28) ÷ (2 x-4) = 4 x^2+6+(-4)/(2 x-4) (Simplify your answers. Do not factor. Use integers or fractions for any numbers in the expressions.)$

Transcript text: Divide using long division. State the quotient, $\mathrm{q}(\mathrm{x})$, and the remainder, $\mathrm{r}(\mathrm{x})$. \[ \begin{array}{l} \left(8 x^{3}-2 x^{2}-14 x-28\right) \div(2 x-4) \\ \left(8 x^{3}-2 x^{2}-14 x-28\right) \div(2 x-4)=4 x^{2}+6+\frac{-4}{2 x-4} \end{array} \] (Simplify your answers. Do not factor. Use integers or fractions for any numbers in the expressions.)

Solution

Solution Steps

To solve the polynomial division problem using long division, we need to divide the polynomial $8x^3 - 2x^2 - 14x - 28$ by the binomial $2x - 4$. The process involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by this result, subtracting from the original polynomial, and repeating the process with the new polynomial formed. The quotient will be the result of these divisions, and any leftover term will be the remainder.

Step 1: Define the Dividend and Divisor

We start with the polynomial dividend $8x^3 - 2x^2 - 14x - 28$ and the divisor $2x - 4$.

Step 2: Perform Polynomial Division

Using polynomial long division, we divide $8x^3 - 2x^2 - 14x - 28$ by $2x - 4$.

Divide the leading term of the dividend $8x^3$ by the leading term of the divisor $2x$ to get $4x^2$.
Multiply the entire divisor $2x - 4$ by $4x^2$ to get $8x^3 - 16x^2$.
Subtract this from the original polynomial: \[ (8x^3 - 2x^2 - 14x - 28) - (8x^3 - 16x^2) = 14x^2 - 14x - 28 \]
Repeat the process with the new polynomial $14x^2 - 14x - 28$:
- Divide $14x^2$ by $2x$ to get $7x$.
- Multiply $2x - 4$ by $7x$ to get $14x^2 - 28x$.
- Subtract: \[ (14x^2 - 14x - 28) - (14x^2 - 28x) = 14x - 28 \]
Repeat again with $14x - 28$:
- Divide $14x$ by $2x$ to get $7$.
- Multiply $2x - 4$ by $7$ to get $14x - 28$.
- Subtract: \[ (14x - 28) - (14x - 28) = 0 \]