Using the Binomial Theorem to Find a Single Term

Expanding a binomial with a high exponent such as (x+2y)16 can be a lengthy process.

Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.

Note the pattern of coefficients in the expansion of (x+y)5.

(x+y)5=x5+(51)x4y+(52)x3y2+(53)x2y3+(54)xy4+y5

The second term is (51)x4y. The third term is (52)x3y2. We can generalize this result.

(nr)xnryr

A General Note: The (r+1)th Term of a Binomial Expansion

The (r+1)th term of the binomial expansion of (x+y)n is:

(nr)xnryr

How To: Given a binomial, write a specific term without fully expanding.

  1. Determine the value of n according to the exponent.
  2. Determine (r+1).
  3. Determine r.
  4. Replace r in the formula for the (r+1)th term of the binomial expansion.

Example 3: Writing a Given Term of a Binomial Expansion

Find the tenth term of (x+2y)16 without fully expanding the binomial.

Solution

Because we are looking for the tenth term, r+1=10, we will use r=9 in our calculations.

(nr)xnryr
(169)x169(2y)9=5,857,280x7y9

Try It 3

Find the sixth term of (3xy)9 without fully expanding the binomial.

Solution