if the binomial (x+y)^7 were expanded, what would be the coefficient of each term?
Accepted Solution
A:
Answer:Term coefficient x⁷ 1x⁶y 7x⁵y² 21x⁴y³ 35x³y⁴ 35x²y⁵ 21xy⁶ 7y⁷ 1Explanation:You can use Pascal's triangle to predict the coefficient of each term in a binomial expansion.Since the binomial has exponent 7, the expanded expression will have 8 terms: (x + y)⁰ has 1 term, (x + y)¹ has two terms, (x + y)² has three terms, (x + y)³ has four terms, and so on.The Pascal triangle for 8 terms has 8 rows and they are: 1 row 1 1 1 row 2 1 2 1 row 3 1 3 3 1 row 4 1 4 6 4 1 row 5 1 5 10 10 5 1 row 6 1 6 15 20 15 6 1 row 71 7 21 35 35 21 7 1 row 8So, the coefficients, in order, are the numbers from the row 8: 1, 7, 21, 35, 35, 21, 7, and 1.And the terms in order are: x⁷y⁰, x⁶y¹, x⁵y², x⁴y³, x³y⁴, x²y⁵, x¹y⁶, and x⁰y⁷.With that, you can write the coefficient of each term:Term coefficient x⁷y⁰ = x⁷ 1x⁶y¹ = x⁶y 7x⁵y² 21x⁴y³ 35x³y⁴ 35x²y⁵ 21x¹y⁶ = xy⁶ 7x⁰y⁷ = y⁷ 1