use synetic division to divide x^4-2x^3-5x^2+x+2 by x-3

To divide \(x^4 - 2x^3 - 5x^2 + x + 2\) by \(x - 3\) using synthetic division, follow these steps:

### Step 1: Set up the synthetic division
- Write the coefficients of the dividend \(x^4 - 2x^3 - 5x^2 + x + 2\). These are: \([1, -2, -5, 1, 2]\).
- Since we are dividing by \(x - 3\), the root is \(3\), which will be used in synthetic division.

### Step 2: Perform synthetic division
1. **Bring down** the first coefficient (1).
2. **Multiply** the root (3) by the first coefficient (1) and **add** it to the second coefficient:
   \[
   3 \times 1 = 3; \quad -2 + 3 = 1
   \]
3. **Multiply** the root (3) by the new coefficient (1) and **add** it to the next coefficient:
   \[
   3 \times 1 = 3; \quad -5 + 3 = -2
   \]
4. **Multiply** the root (3) by the new coefficient (-2) and **add** it to the next coefficient:
   \[
   3 \times -2 = -6; \quad 1 + (-6) = -5
   \]
5. **Multiply** the root (3) by the new coefficient (-5) and **add** it to the final coefficient:
   \[
   3 \times -5 = -15; \quad 2 + (-15) = -13
   \]

### Step 3: Interpret the result
The numbers obtained from the synthetic division process are the coefficients of the quotient and the remainder:
- Quotient: \(x^3 + x^2 - 2x - 5\)
- Remainder: \(-13\)

Thus, the result of the division is:
\[
\frac{x^4 - 2x^3 - 5x^2 + x + 2}{x - 3} = x^3 + x^2 - 2x - 5 + \frac{-13}{x - 3}
\]

Would you like more details or have any questions?

### Related questions:
1. Can synthetic division be used for divisors with higher degree terms (like \(x^2 - 2\))?
2. What is the difference between synthetic division and long division for polynomials?
3. How do you find the remainder theorem for polynomial division?
4. Can you apply synthetic division to complex roots?
5. How does synthetic division relate to the factor theorem?

### Tip:
Synthetic division is a quick method when dividing by linear factors like \(x - c\), but for non-linear divisors, you'll need polynomial long division.

Explore how to use synthetic division to divide the polynomial x^4 - 2x^3 - 5x^2 + x + 2 by x - 3 in a step-by-step solution. This problem covers key algebraic concepts, including synthetic division and the Remainder Theorem, and is suitable for students in Grades 10-12.

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