Expand the following binomials (x ^2 + y^2)^5

Question

(x²+y²⁾⁵

Answer 1

The best thing to use instead of just using brute force is the formula associated with Pascal's triangle. That is (x+y)n=Σⁿ_k=0(ⁿ_k)x^n-ky^k which will then at the value for 5 produce the coefficients 1 5 10 10 5 1 for the decreasing x and increasing y products.

 

Since we are using x² and y² that means they will count in twos instead of ones Starting at X¹⁰y⁰ which is just x¹⁰ and matching with Pascal's Triangle we are able to receive our answer.

 

(x²+y²)⁵=x¹⁰ + 5x⁸y² + 10x⁶y⁴ +10x⁴y⁶ + 5x²y⁸ + y¹⁰