Personal Bankruptcy Laws For Dummies
Book image
Explore Book Buy On Amazon

The final outcomes of a binomial expansion depend on whether the original monomial had no coefficients or exponents (other than 1) of the variables. To find the expansion of binomials with the theorem in a basic situation, follow these steps:

  1. Write out the binomial expansion by using the binomial theorem, substituting in for the variables where necessary.

    If you need to find the entire expansion for a binomial, you can use the binomial theorem:


    For example, consider the problem (m + 2)4. According to the theorem, you should replace the letter a with m, the letter b with 2, and the exponent n with 4:


    The exponents of m begin at 4 and end at 0. Similarly, the exponents of 2 begin at 0 and end at 4. For each term, the sum of the exponents in the expansion is always 4.

  2. Find the binomial coefficients.

    This example uses the combinations formula to find the five coefficients, but you could use Pascal's triangle as a shortcut because the degree is so low (it wouldn't hurt you to write out five rows of Pascal's triangle — starting with 0 through 4).


    You may have noticed that after you reach the middle of the expansion, the coefficients are a mirror image of the first half. This trick is a time-saver you can employ so you don't need to do all the calculations for

  3. Replace all

  4. with the coefficients from Step 2.

    This step gives you

    1(m)4(2)0 + 4(m)3(2)1 + 6(m)2(2)2 + 4(m)1(2)3 + 1(m)0(2)4

  5. Raise the monomials to the powers specified for each term.

  6. Combine like terms and simplify.

    m4 + 8m3 + 24m2 + 32m + 16

Notice that the coefficients you get in the final answer aren't the binomial coefficients you found in Step 1. This difference is because you must raise each monomial to a power (Step 4), and the constant in the original binomial changed the coefficient of each term.

About This Article

This article is from the book:

About the book authors:

James P. Caher, a practicing attorney with 30 years of experience, is a nationally recognized expert on consumer bankruptcies and authority on the Bankruptcy Abuse Prevention and Consumer Protection Act of 2005. He is the coauthor of Debt Free! Your Guide to Personal Bankruptcy Without Shame, Discharging Marital Obligations in Bankruptcy, and Discharging Credit Card Debts in Bankruptcy. Caher also serves on the editorial board of the American Bankruptcy Institute. John M. Caher is a legal journalist who has written about law and the courts for over 25 years. Currently the Albany bureau chief for the New York Law Journal, Caher previously was state editor and legal affairs reporter for the Times Union of Albany. His legal reportage has won more than two dozen awards, including honors from the American Bar Association, the New York State Bar Association, and the Associated Press. He is the author of King of the Mountain: The Rise, Fall and Redemption of Chief Judge Sol Wachtler, and the principal writer assisting former U.S. Treasury Secretary William E. Simon with his autobiography, A Time for Reflection.

This article can be found in the category: