(a-x)*(b-x)*(c-x)*(d-x)*(e-x)* ... *(z-x) =

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Jul 03, 2024

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Expansion of (a-x)(b-x)(c-x)(d-x)(e-x)...(z-x)

In algebra, expanding the product of multiple binomials can be a daunting task. However, there is a fascinating pattern that emerges when we expand the product of all 26 uppercase English letters, each subtracted by a variable x, i.e., (a-x)(b-x)(c-x)(d-x)(e-x)…(z-x). In this article, we will explore this expansion and discover its remarkable properties.

The Expansion Formula

Let's start by expanding the product of two binomials:

(a-x)(b-x) = a^2 - (a+b)x + x^2

Now, let's expand the product of three binomials:

(a-x)(b-x)(c-x) = a^3 - (a+b+c)x^2 + (ab+ac+bc)x - x^3

Observe the pattern emerging:

• The leading term is a^3, which is the product of the variables a, b, and c.
• The middle terms are the sum of all possible products of two variables, each multiplied by x.
• The last term is x^3, which is the product of x with itself three times.

This pattern continues as we expand the product of more binomials. For instance, expanding the product of four binomials gives:

(a-x)(b-x)(c-x)(d-x) = a^4 - (a+b+c+d)x^3 + (ab+ac+ad+bc+bd+cd)x^2 - (abc+abd+acd+bcd)x + x^4

The General Expansion Formula

After expanding the product of n binomials, we can generalize the pattern to obtain the following formula:

(a-x)(b-x)(c-x)(d-x)...(z-x) =

a^n - (sum of all possible products of n-1 variables)x^(n-1) +

(sum of all possible products of n-2 variables)x^(n-2) - ... +

(-1)^(n-1)(sum of all possible products of 1 variable)x +

(-1)^n x^n

where n is the number of binomials in the product.

Properties and Applications

This expansion formula has several remarkable properties and applications:

• Symmetry: The expansion is symmetric with respect to the variables a, b, c, ..., z, meaning that the formula remains the same if we permute the variables in any way.
• Alternating Signs: The coefficients of the terms in the expansion alternate in sign, starting with a positive sign for the leading term.
• Use in Algebra and Combinatorics: This expansion formula has applications in algebra, combinatorics, and other areas of mathematics, such as the calculation of determinants and the study of symmetric functions.

In conclusion, the expansion of (a-x)(b-x)(c-x)(d-x)(e-x)…(z-x) reveals a beautiful pattern that has far-reaching implications in algebra and combinatorics.