Question

How do you use the Binomial Theorem to expand `(y - 2)^4`?

Answer 1

`(y-2)^4=y^4-8y^3+24y^2-32y+16`

Explanation:

According to binomial theorem,

`(a+b)^n=a^n+n/1a^(n-1)b+(n(n-1))/(1*2)a^(n-2)b^2+(n(n-1)(n-2))/(1*2*3)a^(n-3)b^3+.....+b^n`

Here we will find that the coefficients, such as

`{1.n,(n(n-1))/(1*2),(n(n-1)(n-2))/(1*2*3).................n,1}`

will be as per Pascals triangle and symmetric from left to right and right to left.

As such `(a-b)^4=a^4-4/1a^3b+(4*3)/(1*2)a^2b^2-(4*3*2)/(1*2*3)ab^3+(4*3*2*1)/(1*2*3*4)b^4`

or `(a-b)^4=a^4-4a^3b+6a^2b^2-4ab^3+b^4`

Hence `(y-2)^4=y^4-4y^3*2+6y^2*2^2-4y*2^3+2^4`

or `(y-2)^4=y^4-8y^3+24y^2-32y+16`