(1+x)(1+x+x^2)(1+x+x^2+x^3)

4 min read Jun 16, 2024

Exploring the Pattern: (1+x)(1+x+x^2)(1+x+x^2+x^3)

This expression might seem intimidating at first glance, but it hides a beautiful pattern that makes it easier to understand and manipulate. Let's break it down:

Understanding the Pattern

Notice that each factor in the expression is a geometric series:

(1 + x): This is a simple geometric series with the first term 1 and common ratio x.
(1 + x + x²): This is a geometric series with the first term 1 and common ratio x.
(1 + x + x² + x³): Again, a geometric series with the first term 1 and common ratio x.

The only difference between these series is the number of terms.

Simplifying the Expression

We can exploit this pattern to simplify the entire expression. Here's how:

Geometric Series Formula: Recall that the sum of a finite geometric series is given by:
```
S = a(1 - r^n) / (1 - r)
```
where:
- S is the sum of the series
- a is the first term
- r is the common ratio
- n is the number of terms
Applying the Formula: Let's apply this formula to each of our factors:
- (1 + x) = (1 - x²) / (1 - x) (n = 2)
- (1 + x + x²) = (1 - x³) / (1 - x) (n = 3)
- (1 + x + x² + x³) = (1 - x⁴) / (1 - x) (n = 4)

Multiplication: Now, we can multiply all these simplified factors:

[(1 - x²) / (1 - x)] * [(1 - x³) / (1 - x)] * [(1 - x⁴) / (1 - x)]

Simplifying: This simplifies to:

(1 - x²) (1 - x³) (1 - x⁴) / (1 - x)³

Final Result

Therefore, the simplified expression for (1 + x)(1 + x + x²)(1 + x + x² + x³) is (1 - x²) (1 - x³) (1 - x⁴) / (1 - x)³.

Further Observations

Generalization: We can generalize this pattern for any number of factors. The expression (1 + x)(1 + x + x²)...(1 + x + x² + ... + x^(n-1)) simplifies to (1 - x^n) / (1 - x)^n.
Applications: This pattern is useful in various areas of mathematics, including calculus, linear algebra, and abstract algebra.

By recognizing and utilizing the geometric series pattern, we can efficiently simplify complex expressions and gain deeper insights into their structure.