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Binomial Expansion
How do you square a binomial?

Let’s use as a general binomial, and square it:
Next let's show that this pattern will work for all types of binomials:




There are a few things to notice about the pattern:

  • If there is a constant or coefficient in either term, it is squared along with the variables.


  • The powers variable in the first term of the binomial descend in an orderly fashion.
2nd degree, 1st degree, 0 degree or 4th degree, 2nd degree, 0 degree
  • The powers of the variable in the second term ascend in an orderly fashion.
0 degree, 1st degree, 2nd degree
  • The sign of the 2nd term is negative in the 3rd example, as it should be.


  • The sum of the exponents for every term in the expansion is 2.


  • There are 3 terms in the 2nd power expansion.


What if we cube a binomial?




There are a few things to notice about the pattern:

  • If there is a constant or coefficient in either term, it is raised to the appropriate power along with the variables.


  • The powers of the variable in the first term of the binomial descend in an orderly fashion.
3rd degree, 2nd degree, 1st degree, 0 degree
  • powers of the variable in the second term ascend in an orderly fashion.
0 degree, 1st degree, 2nd degree, 3rd degree
  • The signs of the 2nd and 4th term are appropriately negative in the 2nd example.


  • The sum of the exponents in each term of the expansion are 3.


  • There are 4 terms in the 3rd degree expansion.


Summarizing: What patterns do we need to do any binomial expansion?

  • The powers of the first term (the “a” term) descend in consecutive order , starting with the power of the expansion and ending with the zero power . Note that we raise the entire term to that power, then one lower, etc.


  • The powers of the second term (the “b” term) ascend in consecutive integer order, starting with zero power and ending with the power of the expansion.


  • The sum of the exponents (before simplifying them) of each term is the same as the power of the expansion


  • You will always have one more term than the number of the expansion.


  • The signs of an will alternate positive, then negative, etc.
  • The pattern of the coefficients follows Pascal’s Triangle:


Let's Practice:
  1. Expand
Start with the first term


Note there is no need to show the because it is 1
The 2nd term will be


Note that the exponents add up to 5,
The 3rd term will be
The 4th term will be
The 5th term will be
The last term will be


Again there is no need to show the because it is 1.
Therefore
  1. Expand
The degree is 5 so we will have six terms altogether. The coefficients needed to complete the expansion are the 1 5 10 10 5 1 row of Pascal’s Triangle.

Start with the first term


As usual, there is no need to show the because it is 1.
The 2nd term will be


Note that the exponents add up to 5.
The 3rd term will be
The 4th term will be
The 5th term will be
The last term will be
All together we get:

Examples
Expand the following binomial powers.
Example
What is your answer?
 
Example
What is your answer?
 
Example
What is your answer?
 

Things to Think About:
  • Why do the numbers from Pascal’s Triangle (which is a visual array created by Blaise Pascal circa 1623 to 1662) become the coefficients of each term for our expansion?


  • The numbers in each row of Pascal’s Triangle are created by adding the closest two terms of the preceding row. How do the terms of the previous expansion relate to the next expansion (how do the terms of affect the terms of )?


  • How can I find these coefficients without using Pascal’s Triangle? That is, how can I find the specific coefficient for a specific term in any expansion?
These and other questions will be explored in another lesson.


K Mathis

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