In high school, I learned and memorized how to expand the binomial of power 2 and 3. From what I remember, we expand it by multiply term by term and simplify it to the final answer. But there is a simple way can expand it using pascal triangle. Now I’m about to show you how to create and use pascal triangle for binomial expansion.
Here is the example of pascal triangle.
We don’t need to memorize all the number in the pascal triangle, we just need to memorize how to create this.
If we look at this pascal triangle, we can see the first and the last numbers in every rows are alway 1. So, how to get the rest of the numbers?
1st row: 1
2nd row: 1 1
3rd row: 1 1+1=2 1 (add 2 number in 2nd row)
4th row: 1 1+2=3 2+1 = 3 1
5th row: 1 1+3 = 4 3+3=6 3+1=4 1
6th row: 1 1+4 = 5 4+6=10 6+4=10 4+1=5 1
and so on.
For example, in row 6. The first number and the last number is alway equal 1 like I mentioned above.We add first number and 2nd number in row 5 to get the 2nd number in row 6, which is number 5.
Add 2nd number and 3rd number in row 5 to get the 3rd number in row 6, (10).
Add the 3rd number and 4th number in row 5 to get the next number (10).
Add 4th and 5th number to get the next number (5); there is no other number we can add in row 5, so just end the 6th row with number 1. Every other rows are followed with same rules.
This is some example of binomials expansion:
The coefficients of the binomial expansions are the number from pascal triangle. For the power, we can see that
The power of “x” decreased by one in each successive term.
The power of “y” increased by one in each success term.
If we know the rule, we can expand higher powers of the binomial without memorized or multiply term by term.
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