Is $$ x^2+3x-1\text{ a factor of }x^3+2x^2-5x-6 $$

To determine if $x^2 + 3x - 1$ is a factor of $x^3 + 2x^2 - 5x - 6$, we can perform polynomial division or use synthetic division.

### Step 1: Set up the division
We will divide $x^3 + 2x^2 - 5x - 6$ by $x^2 + 3x - 1$.

### Step 2: Perform the division

1. Divide the leading term of the dividend $x^3$ by the leading term of the divisor $x^2$:
   \[
   \frac{x^3}{x^2} = x
   \]
   So, the first term of the quotient is $x$.

2. Multiply the entire divisor $x^2 + 3x - 1$ by $x$ and subtract it from the original polynomial:
   \[
   (x^2 + 3x - 1) \times x = x^3 + 3x^2 - x
   \]
   Subtract this from the original polynomial:
   \[
   (x^3 + 2x^2 - 5x - 6) - (x^3 + 3x^2 - x) = -x^2 - 4x - 6
   \]

3. Repeat the process with the new polynomial $-x^2 - 4x - 6$:
   Divide the leading term $-x^2$ by the leading term $x^2$:
   \[
   \frac{-x^2}{x^2} = -1
   \]
   So, the next term in the quotient is $-1$.

4. Multiply the entire divisor $x^2 + 3x - 1$ by $-1$ and subtract:
   \[
   (x^2 + 3x - 1) \times (-1) = -x^2 - 3x + 1
   \]
   Subtract this from $-x^2 - 4x - 6$:
   \[
   (-x^2 - 4x - 6) - (-x^2 - 3x + 1) = -x - 7
   \]

### Step 3: Conclusion
Since the remainder is $-x - 7$, which is not zero, $x^2 + 3x - 1$ is **not** a factor of $x^3 + 2x^2 - 5x - 6$.

Would you like to see a detailed explanation or have any questions about this process?

Here are five related questions for further exploration:
1. How can you use the Remainder Theorem to check if a polynomial is a factor?
2. What is synthetic division, and how does it differ from polynomial division?
3. How would you factor $x^3 + 2x^2 - 5x - 6$ completely?
4. What is the relationship between roots of a polynomial and its factors?
5. Can $x^2 + 3x - 1$ divide another polynomial, like $x^3 + 5x^2 + 2x - 1$?

**Tip:** Always verify your results by multiplying the quotient and the divisor to ensure the division was performed correctly.

Learn how to determine if x^2 + 3x - 1 is a factor of x^3 + 2x^2 - 5x - 6 using polynomial division and synthetic division. Explore the application of the Remainder Theorem in advanced algebra. Suitable for advanced high school students and those interested in polynomial factorization.

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