https://file.aiquickdraw.com/mathbot/file/1726712129332xfbcaahu.jpg

To solve this problem, we will use synthetic division to divide \(2x^3 - 3x^2 - 4x + 7\) by \(x + 3\). The divisor is \(x + 3\), which implies the root is \(-3\). We will apply synthetic division step-by-step and match it with the provided options.

### Steps for Synthetic Division:
1. **Set up the coefficients of the polynomial**: The coefficients of the dividend \(2x^3 - 3x^2 - 4x + 7\) are \(2, -3, -4, 7\).
2. **Root from the divisor**: Since we are dividing by \(x + 3\), we use \(-3\) for the synthetic division.
3. **Perform the synthetic division**:
   - Write down the coefficients \(2, -3, -4, 7\).
   - Begin with the first coefficient (2), and bring it down.
   - Multiply the root \(-3\) by this first coefficient (2), and add the result to the next coefficient.
   - Repeat this process for all the terms.

Let’s go ahead and compute this.

\[
\begin{array}{r|rrrr}
-3 & 2 & -3 & -4 & 7 \\
   &   & -6 & 27 & -69 \\
\hline
   & 2 & -9 & 23 & -62 \\
\end{array}
\]

### Explanation of steps:
- First, bring down the 2.
- Multiply \(-3 \times 2 = -6\), and add to \(-3\), giving \(-9\).
- Multiply \(-3 \times -9 = 27\), and add to \(-4\), giving \(23\).
- Multiply \(-3 \times 23 = -69\), and add to \(7\), giving \(-62\).

The result of the synthetic division is:
\[
2x^2 - 9x + 23 \quad \text{with a remainder of} \ -62
\]

### Matching with the given options:
- The correct synthetic division should have \(2, -9, 23, -62\).
- Looking at the provided options, the first option matches this result exactly.

### Final Answer:
The correct option is the first one.

---

Do you want further details or have any other questions?

Here are 5 related questions:
1. How is synthetic division different from long division of polynomials?
2. What are the applications of synthetic division?
3. Can synthetic division be used if the divisor is not linear?
4. How do you interpret the remainder in synthetic division?
5. How would the division change if we divided by \(x - 3\) instead?

**Tip**: In synthetic division, always verify the divisor's sign, as using the incorrect sign will lead to wrong results!

Which of the following shows the correct illustration when 2x^3 - 3x^2 - 4x + 7 is divided by x + 3 using synthetic division?

Learn how to perform synthetic division for the polynomial 2x^3 - 3x^2 - 4x + 7 when divided by x + 3. This problem covers key algebraic concepts, including polynomial division and the application of synthetic division, suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation