Question 63563
QUESTION:

Use Pascal's Triangle to expand each binomial.
(a-b)^6

ANSWER:

Pascal's Triangle. 

It is a triangle of numbers,  which is given below:


1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1


Each number in the triangle is the sum of two above. For example, the 6 on line 5 is the sum of the pair of 3's above. So the next line is

1, 10 (1 + 9), 45 (9 + 36), 120 (36 + 84), etc.


Study these numbers and see if you can figure what the next line should be before reading on . . .

Your question.


(x-y)^6

Seventh raw of the triangle gives,COEFFIECIENTS OF THE EXPANSION.

That is, 1 6 15 20 15 6 1

And the variable terms  will come like this,

x^6

x^5 * y^1

x^4* y^2

x^3 * y^3

x^2 * y^4

x^1 * y^5


y^6

(power of the first term will decrease while power of second term will increase till it is 6.)

Since here it is (x - 6), the sign of the term will be alternatively positive and negative like this,

+, -, + , -, +, -, +, - , +, ...................


(x-y)^6=   1x^6+ 6x^5 y^1+ 15x^4 y^2+20x^3 y^3+ 15x^2 y^4+ 6x^1 y^5+ 1y^6

which is the  answer to your question.


Hope you understood.

Regards.

praseenakos@yahoo.co.in