Polynomial Long Division Simplified: Why and How to Divide x2 - 1 by x - 1

Introduction to Polynomial Division

Polynomial division is a method used to divide one polynomial by another. One common technique is the use of long division, which can be applied to various polynomials, including the simplified polynomial x2 - 1 by x - 1. This article will explore the rationale behind this division method and its practical applications.

The Importance of Adding Terms in Polynomial Division

When performing polynomial long division, such as x2 - 1 divided by x - 1, it is necessary to include terms that might seem redundant, such as adding zeros when writing numbers in expanded form. For example, the number 64 is written as 6004 (6 thousands, 0 hundreds, 0 tens, 4 ones) in expanded form. Similarly, in polynomial division, we write x2 - 1 as x2 - - 1 to ensure that all possible degrees of x are represented.

Why Do We Need to Add Terms in Polynomial Division?

In the division process, the term for the x1 (or x) terms is necessary to perform the long division algorithm. Here's an example of how the term is used in the division of 207 by 3. Instead of writing 207 as 207, it is written as 207 (2 hundreds, 0 tens, 7 ones). Similarly, in x2 - 1, writing it as x2 - - 1 helps in the long division process.

Example of Polynomial Long Division

Let's consider the polynomial long division of x2 - 1 by x - 1:

x2 - 1 x x2 - 2x - 1 x x2 - 2x - 4 - 3 (x - 2) (x 1)

Here's a step-by-step breakdown of the division: 1. Divide the leading term of the dividend, x2, by the leading term of the divisor, x, to get x. 2. Multiply the entire divisor, x - 1, by x to get x2 - x. 3. Subtract x2 - x from the original polynomial x2 - 1 to get x - 1. 4. Repeat the process by dividing x (the next term) by x to get 1. 5. Multiply the entire divisor by 1 to get x - 1. 6. Subtract x - 1 from x - 1 to get 0. Thus, the quotient is x 1, and the remainder is 0. The expression can be written as (x - 1) (x 1) x2 - 1, confirming the division.

Understanding Without Tabulation

Another interesting way to understand polynomial division is by Mr. Yaron's approach. Let's look at a more complex example, the division of x8 - 1 by x - 1:

x8 - 1 (x - 1) (x7 x6 x5 x4 x3 x2 x 1) This can be derived through synthetic division or by recognizing the geometric series sum formula. The steps involve subtracting the terms of the divisor from the dividend step-by-step. Mathematically, it can be represented as:

x8 - (-2)8 (x - (-2)) (x7 - 2x6 4x5 - 8x4 16x3 - 32x2 64x - 128)

By summing the series, you get the polynomial quotient.

Conclusion

Polynomial long division, whether using a tabular method or alternative approaches, is a powerful tool in algebra. Understanding the rationale behind adding terms and the process itself helps in performing these operations accurately. By practicing with different polynomials, students can gain a deeper understanding of algebraic manipulation and division techniques.

References

1. Algebra I For Dummies by Mary Jane Sterling. 2. Long Division in Algebra by Example, K. I. Cutler. 3. Polynomial Division Using Geometric Series Formula by A. Yaron.