Question

If the binomial (x+y)^7 were expanded, what would be the coefficient of each term?

Answer 1

Term coefficient

• x⁷ 1

• x⁶y 7

• x⁵y² 21

• x⁴y³ 35

• x³y⁴ 35

• x²y⁵ 21

• xy⁶ 7

• y⁷ 1

Step-by-step explanation:

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 7

1 7 21 35 35 21 7 1 row 8

So, 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 1

x⁶y¹ = x⁶y 7

x⁵y² 21

x⁴y³ 35

x³y⁴ 35

x²y⁵ 21

x¹y⁶ = xy⁶ 7

x⁰y⁷ = y⁷ 1