To preform the division x-2/x^2 +3x+3, begin writing the synthetic division problem blank/blank 3 3

To perform the division \(\frac{x-2}{x^2 + 3x + 3}\) using synthetic division, we need to rewrite this expression and analyze it step-by-step. However, synthetic division generally applies to dividing a polynomial by a linear divisor (like \(x - c\)). Since the divisor here, \(x^2 + 3x + 3\), is quadratic, synthetic division can't be directly applied. Instead, we can explore using polynomial long division or factoring methods if applicable.

Here’s how we could start with polynomial long division:

### Step 1: Set up the division
We are dividing \(x - 2\) by \(x^2 + 3x + 3\), and we'll apply polynomial long division. The setup looks like this:

\[
\frac{x - 2}{x^2 + 3x + 3}
\]

Since the degree of the denominator is higher than the degree of the numerator, the result will be a fraction where no further division can simplify the expression directly. Therefore, synthetic division is not applicable here.

Would you like to go through polynomial long division or try factoring the quadratic?

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Here are some questions related to division and polynomial operations:

1. How can we factor quadratic expressions like \(x^2 + 3x + 3\)?
2. What is the difference between synthetic division and polynomial long division?
3. When is synthetic division useful?
4. How do we handle division when the degree of the numerator is less than the degree of the denominator?
5. What other methods are available for simplifying rational expressions?

**Tip:** When dividing polynomials, synthetic division is faster for linear divisors, but polynomial long division works universally.

Learn how to approach the division of rational expressions, specifically the division of x-2 by x^2 + 3x + 3. This problem involves polynomial division and understanding when synthetic division or polynomial long division applies. Suitable for Grades 10-12 students exploring algebraic concepts.

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