divide the following polynomial using the synthetic division. 1.) (2x³ - x² - 12x - 9) ÷ (x - 3)

To perform synthetic division on the polynomial \( 2x^3 - x^2 - 12x - 9 \) by the divisor \( x - 3 \), follow these steps:

### Step 1: Set up the synthetic division.
- Write down the coefficients of the polynomial: \( 2, -1, -12, -9 \).
- Write the zero of the divisor \( x - 3 \) which is \( 3 \).

### Step 2: Perform the synthetic division.
1. **Bring down** the first coefficient \( 2 \) directly below the line.
2. **Multiply** the divisor's zero \( 3 \) by the value just written below the line \( 2 \), which gives \( 6 \).
3. **Add** this result to the next coefficient \( -1 \), resulting in \( -1 + 6 = 5 \).
4. **Multiply** \( 5 \) by \( 3 \) to get \( 15 \), and then add this to the next coefficient \( -12 \), giving \( -12 + 15 = 3 \).
5. **Multiply** \( 3 \) by \( 3 \) to get \( 9 \), and then add this to the final coefficient \( -9 \), giving \( -9 + 9 = 0 \).

### Step 3: Interpret the result.
- The bottom row now represents the coefficients of the quotient and the remainder:
  - The quotient is \( 2x^2 + 5x + 3 \).
  - The remainder is \( 0 \).

### Conclusion:
The division of \( 2x^3 - x^2 - 12x - 9 \) by \( x - 3 \) yields:

\[
\frac{2x^3 - x^2 - 12x - 9}{x - 3} = 2x^2 + 5x + 3
\]

The remainder is \( 0 \), indicating that \( x - 3 \) is a factor of the polynomial.

Would you like further details on any part of this process? Here are some related questions:

1. How do you factor polynomials using synthetic division?
2. What is the difference between synthetic division and long division of polynomials?
3. How do you find the roots of the polynomial after dividing?
4. Can synthetic division be used for dividing by polynomials of degree higher than 1?
5. What happens if the remainder is not zero in synthetic division?

**Tip:** Synthetic division is a quick method for dividing polynomials, but it only works when dividing by a linear polynomial of the form \( x - c \).

Explore the process of dividing the polynomial 2x³ - x² - 12x - 9 by x - 3 using synthetic division. This detailed step-by-step solution covers key algebraic concepts such as polynomial division, synthetic division, and the application of the Remainder and Factor Theorems. Ideal for students in Grades 10-12.

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