(7+x)^3

4 min read Jun 16, 2024

Expanding (7+x)^3

The expression (7+x)^3 represents the cube of the binomial (7+x). Expanding this expression involves multiplying the binomial by itself three times. There are a few ways to approach this:

1. Direct Multiplication

This method involves multiplying the binomial by itself three times in a step-by-step process.

(7+x)^3 = (7+x)(7+x)(7+x)

Expand the first two terms: (7+x)(7+x) = 49 + 14x + x^2
Multiply the result by (7+x): (49 + 14x + x^2)(7+x) = 343 + 98x + 7x^2 + 98x + 14x^2 + x^3
Combine like terms: (7+x)^3 = x^3 + 21x^2 + 196x + 343

2. Binomial Theorem

The Binomial Theorem provides a formula for expanding any binomial raised to a power. The formula states:

(a+b)^n = a^n + (n choose 1)a^(n-1)b + (n choose 2)a^(n-2)b^2 + ... + (n choose n-1)ab^(n-1) + b^n

where (n choose k) represents the binomial coefficient, calculated as:

(n choose k) = n! / (k! * (n-k)!)

Applying this to our case:

(7+x)^3 = 7^3 + (3 choose 1)7^2x + (3 choose 2)7x^2 + x^3

Calculating the binomial coefficients:

(3 choose 1) = 3!/(1! * 2!) = 3 (3 choose 2) = 3!/(2! * 1!) = 3

Substituting back:

(7+x)^3 = 343 + 3 * 49x + 3 * 7x^2 + x^3

Simplifying:

**(7+x)^3 = x^3 + 21x^2 + 196x + 343

3. Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The numbers in each row correspond to the binomial coefficients.

             1
            1 1
           1 2 1
          1 3 3 1
         1 4 6 4 1
        1 5 10 10 5 1
       1 6 15 20 15 6 1
      1 7 21 35 35 21 7 1
     ............................

The coefficients for (7+x)^3 are found in the fourth row of Pascal's Triangle: 1, 3, 3, 1.

Using these coefficients, we can expand the expression:

(7+x)^3 = 1 * 7^3 + 3 * 7^2 * x + 3 * 7 * x^2 + 1 * x^3