Expanding binomial expression using JavaScript

We need to write a JavaScript function that expands binomial expressions of the form (ax+b)^n, where a and b are integers, x is a variable, and n is a natural number. The function returns the expanded polynomial as a string.

Problem Statement

Given an expression like (8a+6)^4, we need to expand it using the binomial theorem. The result should be in the form ax^b+cx^d+ex^f... with coefficients and powers in decreasing order.

Understanding the Binomial Theorem

The binomial theorem states that (x+y)^n = ?(C(n,k) * x^(n-k) * y^k) where C(n,k) is the binomial coefficient.

Implementation

const str = '(8a+6)^4';

const trim = value => value === 1 ? '' : value === -1 ? '-' : value;

const factorial = (value, total = 1) =>
    value <= 1 ? total : factorial(value - 1, total * value);

const find = (str = '') => {
    let [op1, coefficient, variable, op2, constant, power] = str
        .match(/(\W)(\d*)(\w)(\W)(\d+)..(\d+)/)
        .slice(1);
    
    power = +power;
    
    if (!power) {
        return '1';
    }
    
    if (power === 1) {
        return str.match(/\((.*)\)/)[1];
    }
    
    coefficient = op1 === '-' 
        ? coefficient 
            ? -coefficient 
            : -1
        : coefficient 
            ? +coefficient 
            : 1;
    
    constant = op2 === '-' ? -constant : +constant;
    
    const factorials = Array.from({ length: power + 1 }, (_, i) => factorial(i));
    let result = '';
    
    for (let i = 0, p = power; i <= power; ++i, p = power - i) {
        let judge = factorials[power] / (factorials[i] * factorials[p]) * 
                   (coefficient ** p) * (constant ** i);
        
        if (!judge) {
            continue;
        }
        
        result += p 
            ? trim(judge) + variable + (p === 1 ? '' : `^${p}`)
            : judge;
        result += '+';
    }
    
    return result.replace(/\+\-/g, '-').replace(/\+$/, '');
};

console.log(find(str));

4096a^4+6144a^3+3456a^2+864a+81

How It Works

The function uses regular expressions to parse the input expression, extracting the coefficient, variable, constant, and power. It then applies the binomial theorem using precomputed factorials for efficiency.

Key Components

Example Breakdown

For (8a+6)^4:

• Coefficient a = 8, constant b = 6, power n = 4
• Each term follows: C(4,k) * (8a)^(4-k) * 6^k
• Results in 5 terms with decreasing powers of a

Conclusion

This implementation efficiently expands binomial expressions using the mathematical binomial theorem. The function handles coefficient formatting and produces clean polynomial output in standard mathematical notation.