BINOMIAL EXPANSION
(x+3)² can be solved within a few calculations, but what about when you raise the expression to the power of 5? (x+3)⁷? (x+3)¹¹?
Pascal's triangle can help us!
Pascal's triangle can help us!
Example:
The expansion of (x+y)⁷ 1. Looking at row 7 of Pascal's triangle, we know our coefficients are 1, 7, 21, 35, 35, 21, 7, 1 2. Each term is going to be a combination of x and y powers. We know that our first term will just contain an x and our last term will be a y. Writing out the expansion with just coefficients, x's and y's: |
(x + y)⁷ = x +7xy + 21xy + 35xy + 35xy + 21xy +7xy + y
There's still something missing..the exponents!
3. (x+y)⁷ = x⁷ + 7x⁶y¹ + 21x⁵y² + 35x⁴y³ + 35x³y⁴ + 21x²y⁵ + 7x¹y⁶ + y⁷
The exponents of x and y are set up so that each term produces the sum of 7.
As you can see, the expansion is symmetrical on both sides with the exponents of x and y inversed.
This method works with any binomial expansion!
There's still something missing..the exponents!
3. (x+y)⁷ = x⁷ + 7x⁶y¹ + 21x⁵y² + 35x⁴y³ + 35x³y⁴ + 21x²y⁵ + 7x¹y⁶ + y⁷
The exponents of x and y are set up so that each term produces the sum of 7.
As you can see, the expansion is symmetrical on both sides with the exponents of x and y inversed.
This method works with any binomial expansion!