Question

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

Answer 1

Answer:

Term               coefficient

• x⁷                       1

• x⁶y                     7

• x⁵y²                  21

• x⁴y³                  35

• x³y⁴                  35

• x²y⁵                  21

• xy⁶                     7

• y⁷                      1

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