Expanding a binomial with a high exponent such as 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 .
The second term is . The third term is . We can generalize this result.
A General Note: The (r+1)th Term of a Binomial Expansion
The term of the binomial expansion of is:
How To: Given a binomial, write a specific term without fully expanding.
- Determine the value of according to the exponent.
- Determine .
- Determine .
- Replace in the formula for the term of the binomial expansion.
Example 3: Writing a Given Term of a Binomial Expansion
Find the tenth term of without fully expanding the binomial.
Solution
Because we are looking for the tenth term, , we will use in our calculations.
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