Question

How do you expand the binomial `(3x^2-3)^4` using the binomial theorem?

Answer 1

`(3x^2-3)^4=81x^8-324x^6+486x^4-324x^2+81`

Explanation:

We can use Pascal's triangle to give us the binomial coefficients:
`color(white)(aaaaaaaaaaaaaaaaaaa)1`
`color(white)(aaaaaaaaaaaaaaaaa)1color(white)(aaa)1`
`color(white)(aaaaaaaaaaaaaaa)1color(white)(aaa)2color(white)(aaa)1`
`color(white)(aaaaaaaaaaaaa)1color(white)(aaa)3color(white)(aaa)3color(white)(aaa)1`
`color(white)(aaaaaaaaaaa)1color(white)(aaa)4color(white)(aaa)6color(white)(aaa)4color(white)(aaa)1`

Looking at the 5th row (remember, we're counting from `0`), we can see that the coefficients of the expansion will be `1,4,6,4,1`.

According to the Binomial Theorem, the degree of the first term will decrease from left to right and the degree of the second term will increase from left to right.

We can use this information to construct a formula for the expansion like so:
`(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4`

We can then apply it to our expression:
`(3x^2-3)^4=(3x^2)^4+4(3x^2)^3(-3)+6(3x^2)^2(-3)^2+4(3x^2)(-3)^3+(-3)^4=`

`=81x^8-12(27x^6)+6(9x^4)(9)+12x^2(-27)+81=`

`=81x^8-324x^6+486x^4-324x^2+81`