https://file.aiquickdraw.com/mathbot/file/1729538938878sv6qdjtq.jpg

To solve the given problem using synthetic division, we are tasked with simplifying the expression \(\frac{2x^3 + 8x^2 - 16}{x + 1}\). Here's the step-by-step approach:

### Step 1: Set up the synthetic division
The divisor is \(x + 1\), so we use \(-1\) for the synthetic division. The coefficients of the dividend \(2x^3 + 8x^2 + 0x - 16\) are \(2, 8, 0, -16\).

### Step 2: Perform synthetic division
Set up the synthetic division table:

\[
\begin{array}{r|rrrr}
-1 & 2 & 8 & 0 & -16 \\
   &   & -2 & -6 & 6 \\
\hline
   & 2 & 6 & -6 & -10 \\
\end{array}
\]

Here's how the process works:
1. Bring down the 2.
2. Multiply \(-1\) by 2 (which is \(-2\)) and add it to the next coefficient 8, giving \(6\).
3. Multiply \(-1\) by 6 (which is \(-6\)) and add it to 0, giving \(-6\).
4. Multiply \(-1\) by \(-6\) (which is \(6\)) and add it to \(-16\), giving \(-10\).

### Step 3: Write the quotient and remainder
The result of the synthetic division is:
- Quotient: \(2x^2 + 6x - 6\)
- Remainder: \(-10\)

Thus, the expression becomes:
\[
\frac{2x^3 + 8x^2 - 16}{x + 1} = 2x^2 + 6x - 6 + \frac{-10}{x + 1}
\]

This is in the required form \(q(x) + \frac{r}{d(x)}\), where:
- \(q(x) = 2x^2 + 6x - 6\)
- \(r = -10\)
- \(d(x) = x + 1\)

Would you like more details or have any questions about this process?

### Related questions:
1. How is synthetic division different from long division of polynomials?
2. What are the advantages of using synthetic division over traditional methods?
3. Can synthetic division be used for divisors of the form \(x - a\) where \(a \neq 1\)?
4. How do you handle synthetic division when the divisor has a coefficient other than 1 for \(x\)?
5. How can you verify the result of synthetic division using polynomial multiplication?

### Tip:
Always check your synthetic division setup carefully—especially the signs of coefficients and terms, as a small mistake can lead to an incorrect result.

Use synthetic division to simplify \(\frac{2x^3 + 8x^2 - 16}{x + 1}\). Write your answer in the form \(q(x) + \frac{r}{d(x)}\), where \(q(x)\) is a polynomial, \(r\) is an integer, and \(d(x)\) is a linear polynomial. Simplify any fractions.

Learn how to simplify the polynomial expression \(\frac{2x^3 + 8x^2 - 16}{x + 1}\) using synthetic division. This step-by-step solution explains key algebraic concepts such as synthetic division and polynomial division. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation