Question 338972: Please help me solve this equation:
determine the second and fifth terms in the expanison of (2x-3y)^5?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
 Start with the given expression
To expand this, we're going to use binomial expansion. So let's look at Pascal's triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Looking at the row that starts with 1,5, etc, we can see that this row has the numbers:
1, 5, 10, 10, 5, and 1
These numbers will be the coefficients of our expansion. So to expand , simply follow this procedure:
Write the first coefficient. Multiply that coefficient with the first binomial term  and then the second binomial term  . Repeat this until all of the coefficients have been written.
Once that has been done, add up the terms like this:
 Notice how the coefficients are in front of each term.
However, we're not done yet.
 Looking at the first term  , raise to the 5th power and raise to the 0th power.
 Looking at the second term  raise to the 4th power and raise to the 1st power.
Continue this until you reach the final term.
Notice how the exponents of are stepping down and the exponents of are stepping up.
So the fully expanded expression should now look like this:
 Distribute the exponents
 Multiply
 Multiply the terms with their coefficients
So expands and simplifies to .
In other words, 
From the last equation, we can see that the 2nd term is  and the 5th term is 
Shortcut: You can find the kth term of the expansion  by using the formula  where  is the coefficient (either found with a calculator or by using Pascal's triangle).
If you need more help, email me at jim_thompson5910@hotmail.com
Also, feel free to check out my tutoring website.
Jim
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