Understanding the Polynomial Division of x^2 - x - 12 ÷ x - 6

In this article, we will explore the process of polynomial division using the example (frac{x^2 - x - 12}{x - 6}). We will cover both the long division method and the synthetic division method to find the quotient and the remainder. This article is relevant for students, educators, and anyone interested in understanding polynomial division.

Polynomial Division Methods

Polynomial division can be performed using either long division or synthetic division. Each method has its own advantages and is suitable in different scenarios. In this article, we will demonstrate both methods using the given polynomial (frac{x^2 - x - 12}{x - 6}).

Long Division Method

Step 1: Set up the division.

Set up the long division by writing the divisor (x - 6) outside the division symbol and the dividend (x^2 - x - 12) inside.

Step 2: Perform the division using long division.

x - 6
--------------------------------
x - 6 | x^2 - x - 12
      - (x^2 - 6x)
      ----------------
           5x - 12
           - (5x - 30)
           ------------
                42

Step 3: Analyze the result.

The division results in a quotient of (x 5) and a remainder of (42). We can express this result as:

x^2 - x - 12  (x - 6)(x   5)   42

Verifying the Division

To verify the result, we can multiply the quotient by the divisor and add the remainder:

(x - 6)(x   5)   42 x^2   5x - 6x - 30   42 x^2 - x - 12

Thus, the division is correct.

Synthetic Division Method

Step 1: Identify the coefficients of the dividend and the divisor.

The coefficients of the dividend (x^2 - x - 12) are (1, -1, -12). The root of the divisor (x - 6) is (6).

Step 2: Perform synthetic division.

6 | 1 -1 -12  | 6  30  ------------    1  5  42

Step 3: Interpret the result.

The result of synthetic division gives us the coefficients of the quotient (x 5) and the remainder (42). Thus, we can write:

x^2 - x - 12  (x - 6)(x   5)   42

Conclusion

Both long division and synthetic division methods are effective for dividing polynomials. In this example, we demonstrated the long division method and the synthetic division method, both yielding:

(x^2 - x - 12 (x - 6)(x 5) 42).

Understanding these methods is essential for solving more complex polynomial problems and for further topics in algebra and calculus. Whether you are a student, educator, or a professional, mastering these techniques is valuable.