(x+1)(x+2)(x+3)(x+4)(x+5)

4 min read Jun 16, 2024

Exploring the Expansion of (x+1)(x+2)(x+3)(x+4)(x+5)

This seemingly simple expression hides a fascinating pattern and can be expanded in multiple ways. Let's delve into the different approaches and explore the resulting polynomial.

Direct Expansion:

The most straightforward approach is to expand the expression term by term.

Multiply the first two factors: (x+1)(x+2) = x² + 3x + 2
Multiply the result by the third factor: (x² + 3x + 2)(x+3) = x³ + 6x² + 11x + 6
Continue multiplying by the remaining factors: (x³ + 6x² + 11x + 6)(x+4) = x⁴ + 10x³ + 35x² + 50x + 24
Finally, multiply by the last factor: (x⁴ + 10x³ + 35x² + 50x + 24)(x+5) = x⁵ + 15x⁴ + 85x³ + 225x² + 274x + 120

While this method works, it can become tedious for larger expressions.

Utilizing Patterns:

Observe that the expression represents the product of five consecutive integers increased by 1. This leads to interesting patterns:

The constant term: The product of the constant terms in each factor (12345) equals 120. This is always the product of the consecutive integers.
The leading coefficient: The leading coefficient is always 1, as the highest power of x is obtained by multiplying the 'x' terms from each factor.
The coefficients of other terms: The coefficients of the other terms are related to the sum of products of the consecutive integers. For example, the coefficient of x⁴ is the sum of all possible products of four of the consecutive integers (1234 + 1235 + 1245 + 1345 + 234*5), which equals 15.

The Binomial Theorem Approach:

While less obvious, the Binomial Theorem can be used to expand this expression. We can rewrite the expression as:

(x+1)(x+2)(x+3)(x+4)(x+5) = (x+5) * (x+4) * (x+3) * (x+2) * (x+1)

Now, let's consider the product of the last four factors:

(x+4) * (x+3) * (x+2) * (x+1) = (x⁴ + 10x³ + 35x² + 50x + 24)

This resembles the form of the Binomial Theorem expansion. We can rewrite it as:

(x + 5) * (x⁴ + 10x³ + 35x² + 50x + 24) = (x+5) * (x⁴ + (105)x³ + (355²)x² + (505³)x + (245⁴))

Finally, expanding this product gives us the same result as the direct expansion: x⁵ + 15x⁴ + 85x³ + 225x² + 274x + 120

Conclusion:

Expanding (x+1)(x+2)(x+3)(x+4)(x+5) reveals a fascinating polynomial with interesting patterns. While direct expansion is the most straightforward method, utilizing patterns and even the Binomial Theorem offer alternative approaches to understanding this intriguing expression.