Questions: Question 8 of 14 -11 Divide the polynomials by either long division or synthetic division. (x^3+2x^2-4x-8) / (x+2) Indicate the quotient, Q(x), and the remainder, r(x), in the equation P(x) / d(x) = Q(x) + r(x) / d(x) where P(x) = x^3+2x^2-4x-8 and d(x) = x+2. Q(x) = r(x) =

Question 8 of 14
-11

Divide the polynomials by either long division or synthetic division.

(x^3+2x^2-4x-8) / (x+2)

Indicate the quotient, Q(x), and the remainder, r(x), in the equation

P(x) / d(x) = Q(x) + r(x) / d(x)

where P(x) = x^3+2x^2-4x-8 and d(x) = x+2.

Q(x) =

r(x) =

Transcript text: Question 8 of 14 $-11$ Divide the polynomials by either long division or synthetic division. \[ \left(x^{3}+2 x^{2}-4 x-8\right) \div(x+2) \] Indicate the quotient, $Q(x)$, and the remainder, $r(x)$, in the equation \[ \frac{P(x)}{d(x)}=Q(x)+\frac{r(x)}{d(x)} \] where $P(x)=x^{3}+2 x^{2}-4 x-8$ and $d(x)=x+2$. \[ Q(x)= \] \[ r(x)= \]

Solution

Solution Steps

To solve the problem of dividing the polynomial $ P(x) = x^3 + 2x^2 - 4x - 8 $ by $ d(x) = x + 2 $, we can use synthetic division. Synthetic division is a simplified form of polynomial division, particularly useful when dividing by a linear factor. The steps involve using the root of the divisor (in this case, -2) and performing operations on the coefficients of the dividend polynomial to find the quotient and remainder.

Step 1: Polynomial Division Setup

We are tasked with dividing the polynomial $ P(x) = x^3 + 2x^2 - 4x - 8 $ by the linear polynomial $ d(x) = x + 2 $. We will use synthetic division to perform this operation.

Step 2: Perform Synthetic Division

Using the root of the divisor $ d(x) $, which is $ -2 $, we apply synthetic division to the coefficients of $ P(x) $:

Coefficients of $ P(x) $: $ [1, 2, -4, -8] $
Root: $ -2 $

The synthetic division process yields:

Quotient coefficients: $ [1, 0, -4] $
Remainder: $ 0 $

Step 3: Express the Results

From the synthetic division, we can express the results as follows:

The quotient $ Q(x) $ is given by the polynomial formed from the coefficients: \[ Q(x) = 1x^2 + 0x - 4 = x^2 - 4 \]
The remainder $ r(x) $ is: \[ r(x) = 0 \]