Dividing Polynomials: Dividing 6x^3 - x^2 - 1 - 3 by 2x - 3 Using Long Division

Polynomial division is a fundamental skill in algebra that allows us to simplify complex expressions. In this article, we will walk through the process of dividing the polynomial 6x^3 - x^2 - 1 - 3 by 2x - 3 using the long division method. This method is particularly useful for dividing polynomials of higher degree by linear expressions.

Step-by-Step Guide to Long Division

To divide 6x^3 - x^2 - 1 - 3 by 2x - 3 using long division, follow these steps:

Write down the expression: Start by setting up the problem in long division format. The divisor is written outside, and the dividend (the expression to be divided) is written inside the division symbol. Divide the leading terms: Divide the first term of the dividend by the first term of the divisor. This gives the first term of the quotient. In this case, divide 6x^3 by 2x, which results in 3x^2. Multiply and subtract: Multiply the entire divisor by the first term of the quotient, and subtract this product from the original dividend. This step helps to eliminate the highest degree term of the dividend. Here, subtract (3x^2 * (2x - 3)) 6x^3 - 9x^2 from 6x^3 - x^2 - 1 - 3. This gives (8x^2 - x^2 - 1 - 3) 8x^2 - 1 - 3. Repeat the process: Bring down the next term of the dividend, and repeat the division and subtraction steps until all terms are processed. For this step, combine like terms and continue the process. Check for a remainder: The final result should either be a term that cannot be divided or a remainder that is less in degree than the divisor.

Performing the Long Division

Step 1: Setting up the Division

Write down the problem:

   _3x^2_4x-1__                                                                                                  2x - 3 | 6x^3 - x^2 - 1 - 3

Step 2: Dividing the Leading Terms

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

   _3x^2_4x-1__                                                                                                  2x - 3 | 6x^3 - x^2 - 1 - 3         - 6x^3 - 9x^2         ______________

Subtract this from the original expression:

   _3x^2_4x-1__                                                                                                  2x - 3 | 6x^3 - x^2 - 1 - 3         - 6x^3 - 9x^2         ______________                   8x^2 - x^2 - 1 - 3

Step 3: Continue the Process

Bring down the next terms, and repeat the division and subtraction:

   _3x^2_4x-1__                                                                                                  2x - 3 | 6x^3 - x^2 - 1 - 3         - 6x^3 - 9x^2         ______________                   8x^2 - 8x^2 - 12x

Subtract again:

   _3x^2_4x-1__                                                                                                  2x - 3 | 6x^3 - x^2 - 1 - 3         - 6x^3 - 9x^2         ______________                   8x^2 - 12x - 1 - 3         - 8x^2 - 12x         ______________                                      - 4 - 3         - - 2x - 3         ______________                                  0

The final result is that the quotient is 3x^2 4x - 1 with no remainder.

Conclusion

In conclusion, the division of 6x^3 - x^2 - 1 - 3 by 2x - 3 results in:

6x^3 - x^2 - 1 - 3  (2x - 3)(3x^2   4x - 1)

This method of polynomial division using long division is a powerful tool in algebra, providing a clear and systematic approach to simplifying polynomial expressions.

Keywords

polynomial division long division method synthetic division algebra polynomial

Additional Resources

For more information on polynomial division and other algebraic concepts, explore the following resources:

Algebra Practice Website Math Tutoring Platform

These resources can help you deepen your understanding and practice different methods of polynomial division.