https://file.aiquickdraw.com/mathbot/file/17289620183687x2oepbp.jpg

To solve the given division problem using synthetic division, we'll follow these steps:

We are dividing \( x^3 + 5x^2 + 8x + 6 \) by \( x + 3 \).

### Step-by-Step Synthetic Division:

1. **Set up the synthetic division:**
   - The divisor is \( x + 3 \), so we use \( -3 \) (the opposite sign of 3).
   - The coefficients of the dividend \( x^3 + 5x^2 + 8x + 6 \) are: \( 1, 5, 8, 6 \).

2. **Perform the synthetic division process:**

   We set up the coefficients in a row: \( [1, 5, 8, 6] \), and the divisor \( -3 \).

   - Bring down the first coefficient (1).
   
   - Multiply \( -3 \) by 1 (the value you just brought down) and add it to the next coefficient (5):
     \[
     5 + (-3 \times 1) = 5 - 3 = 2
     \]

   - Multiply \( -3 \) by 2 and add it to the next coefficient (8):
     \[
     8 + (-3 \times 2) = 8 - 6 = 2
     \]

   - Multiply \( -3 \) by 2 and add it to the next coefficient (6):
     \[
     6 + (-3 \times 2) = 6 - 6 = 0
     \]

So, the result of synthetic division gives the following coefficients: \( 1, 2, 2 \), with a remainder of 0.

### Final Answer:
The quotient is \( x^2 + 2x + 2 \), and there is no remainder. Therefore:

\[
\frac{x^3 + 5x^2 + 8x + 6}{x + 3} = x^2 + 2x + 2
\]

Would you like further details or clarification? Here are some additional questions to expand on the topic:

1. What is synthetic division, and how does it differ from long division?
2. How do you interpret a remainder in synthetic division?
3. Can synthetic division be applied when the divisor is not a linear binomial like \(x + 3\)?
4. What are other methods to divide polynomials besides synthetic division?
5. How does synthetic division help in finding roots or factors of polynomials?

**Tip:** When setting up synthetic division, always ensure the dividend is written in descending powers of the variable, filling in any missing terms with a coefficient of 0.

Use synthetic division to perform the division: (x^3 + 5x^2 + 8x + 6) ÷ (x + 3).

Learn how to divide the polynomial x^3 + 5x^2 + 8x + 6 by x + 3 using synthetic division. This detailed guide provides step-by-step instructions for students in Grades 9-12, including key algebraic concepts and the remainder theorem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation