Binomial Expansion Simplified!!

Binomial Expansion!

Ugly? Difficult? Confusing?

None of the above if you let your brain take advantage of the nice symmetry or repeatability.

Most teachers and text books show students to expand a binomial horizontally.

It is much quicker and nicer to expand vertically.

I am going to expand the following vertically and will show each step.




since the power is 4 there will be 5 terms.

steps 1, 2 , and 3: write a column of 5 C's, a column of 5 x's, and a column of 5 3's.

(be sure to write all the C's then all the x's, and then all the 3's, be careful it might even be fun)

      C       x       3

      C       x       3

      C       x       3

      C       x       3

      C       x       3


Step 4: write exponents on the x's starting at the top with the 4 and ending at 0 at the bottom.

      C        x4      3

      C        x3      3

      C        x2      3

      C        x1      3

      C        x0      3


Step 5: write exponents on the 3's starting at the top with 0 and ending at 4 at the bottom.

      C        x4      30

      C        x3      31

      C        x2      32

      C        x1      33

      C       x0      34


Step 6: write the exponent from the binomial, 4, as the "n" on each C.

      4C        x4      30

      4C        x3      31

      4C        x2      32

      4C        x1      33

      4C       x0      34


Step 7: write the exponents from the third column as the "k" on each C.

      4C0       x4      30

      4C1       x3      31

      4C2       x2      32

      4C3       x1      33

      4C4       x0      34


Step 8: work out each column, one colmn at a time.

     1        x4       1

     4        x3       3

     6        x2       9

     4        x        27

     1        1        81


Step 9: multiply across rows to get each term of the expansion.


x4 + 12x3 + 54x2 + 108x + 81


This vertical method if practiced only a few times will always yield the expansion quicker then the horizontal method. 

Be careful if the second term of the binomial is negative. Then you must include that negative in the third column with brackets around the whole term to make sure that the negative and the # both get affected by the exponent.

ie: (-3)3 = -27


This also applies if either term of the binomial has more then one factor involved.

ie: (2x)4 = 16x4


Of course, you won't always be asked to write the entire expansion. You may only be asked to find a certain term of the expansion. The symmetry seen above yields a formula to use in these cases.


Stay tuned for my next BLOG, today as well, to see how to use the formula to find certain terms. 


Dec 13, 2012 Posted by: Tom MacFarlane
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