Question

How do you use pascals triangle to expand `(3a-b)^4`?

Answer 1

`(3a-b)^4=81a^4-108a^3b+54a^2b^2-12ab^3+b^4`

Explanation:

The line of pascals triangle that corresponds to the `(x+y)^4` expansion contains the numbers 1, 4, 6, 4, 1. It is these numbers that we are going to use as our leading coefficients in the expansion process.

As a result the generalised `(x+y)^4` expansion is `x^4+4x^3y+6x^2y^2+4xy^3+y^4`

To complete the expansion for `(3a+b)^4` all we have to do is substitute 3a for x and -b for y and we have;

`(3a)^4+4xx(3a)^3(-b)+6xx(3a)^2(-b)^2+4(3a)(-b)^3+(-b)^4`

Which is equivalent to:

`3^4a^4+4xx3^3a^3(-b)+6xx3^2a^2(-b)^2+4(3a)(-b)^3+(-b)^4`

Evaluating the radicals and combining like terms will then give;

`81a^4-108a^3b+54a^2b^2-12ab^3+b^4`

Hope it helps :)