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Alg2TrigH_-_Notes_9.5_-_2018.pdf
D
IS cont'd
Binomial Expansion 2 of4
Note : There are several observations that should be made about
the pattern and examples just listed .
@ In each expansion, there are 'nt !"terms.
2 'x 'and y " have symmetrical roles. The powers of 'x'decrease
by 1.
by 1 in successive terms, whereas the powers of y "increase
The sum of the powers of each term is "n't For instance
3+124
2+2-4
(xty)"= x'+ 4x'y tory 's try ty ?
The coefficients of each term increase then decrease in
symmetric pattern. This pattern is often called Pascal's
Triangle. it proceeds at follows :
(xty). = 1
7 0 throw
(xty)'- Xty
-> /st row
(xty)* =x'+2xy +y"
-> 2ad row
->1 21
37 row
-1 3 3 1
eh+, hxE + K XE +(X = =(h4x)
(xry)* =x'tyxy + bxay*+ /xy ' ty? -7 41 row ->1 4 6 4 1
This pattern can be continued forever. Note in the above
triangle of numbers how each individual number is able to
be created by adding the two numbers above it (with every
row beginning and ending with U. The next two rows of the-
triangle are shen
Each row of the triangle alto corresponds to an nC, "volue .
For example , taking the 6throw at shown abore :
20
you can verify these by hand or in the calculator. 50 to [MATH ]
then +-- PROP and telect nCr . Fill in the numbers
at needed.
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- Binomial Theorem, ty, st row