Polynomial Long Division: Dividing (X^3 - 4X^2 6X - 4) by (X - 2)

Polynomial long division is a method used to divide one polynomial by another, resulting in a quotient and possibly a remainder. This article will guide you through the detailed process of dividing (X^3 - 4X^2 6X - 4) by (X - 2). Understanding this process is crucial for handling more complex polynomial equations and for applying algebraic concepts effectively.

Setting Up the Division

First, we set up the division as follows:

Here, the dividend is (X^3 - 4X^2 6X - 4), and the divisor is (X - 2).

Dividing the Leading Terms

To begin, we divide the leading term of the dividend by the leading term of the divisor:

(frac{X^3}{X} X^2)

So, the first term of the quotient is (X^2).

Multiplying and Subtracting

Next, we multiply the term (X^2) by the divisor (X - 2), and then subtract this from the original polynomial:

(X^2 cdot (X - 2) X^3 - 2X^2)

Subtracting this from the original polynomial:

((X^3 - 4X^2 6X - 4) - (X^3 - 2X^2) -2X^2 6X - 4)

Repeating the Process

We now divide the leading term of (-2X^2 6X - 4) by (X - 2):

(frac{-2X^2}{X} -2X)

The next term in the quotient is (-2X).

We then multiply (-2X) by the divisor (X - 2):

(-2X cdot (X - 2) -2X^2 4X)

Next, we subtract this product from (-2X^2 6X - 4):

((-2X^2 6X - 4) - (-2X^2 4X) 2X - 4)

Dividing Again

Finally, we divide the leading term of (2X - 4) by (X - 2):

(frac{2X}{X} 2)

The last term in the quotient is (2).

We multiply (2) by the divisor (X - 2):

(2 cdot (X - 2) 2X - 4)

Subtracting this product from (2X - 4):

((2X - 4) - (2X - 4) 0)

Final Result

We can now write the final result of the long division:

(frac{X^3 - 4X^2 6X - 4}{X - 2} X^2 - 2X 2 - frac{8}{X - 2})

This means that the quotient is (X^2 - 2X 2), and the remainder is (-8), with the divisor being (X - 2).

Understanding Polynomial Long Division

This step-by-step process of polynomial long division is quite straightforward:

Divide the leading term of the dividend by the leading term of the divisor.

Multiply the divisor by the quotient term and subtract the result from the original polynomial.

Repeat the process with the remainder until the degree of the remainder is less than the degree of the divisor.

By following these steps, we have explicitly divided (X^3 - 4X^2 6X - 4) by (X - 2), leading to a quotient of (X^2 - 2X 2) and a remainder of (-8).