Question

How do you use the Binomial Theorem to expand `(x + 1)^4`?

Answer 1

`x^4+4x^3+6x^2+4x+1`

Explanation:

The binomial theorem states:
`(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4`

so here, `a=x and b=1`

We get:
`(x+1)^4 = x^4+4x^3(1)+6x^2(1)^2+4x(1)^3+(1)^4`
`(x+1)^4 = x^4+4x^3+6x^2+4x+1`

Answer 2

`1+4x+6x^2+4x^3+x^4`

Explanation:

Binomial expansion is given by:
`(a+bx)^n=sum_(r=0)^n(n!)/(r!(n-r)!)a^(n-r)(bx)^r`

So, for `(1+x)^4` we have:
`(4!)/(0!(4-0)!)1^(4-0)x^0+(4!)/(1!(4-1)!)1^(4-1)x^1+(4!)/(2!(4-2)!)1^(4-2)x^2+(4!)/(3!(4-3)!)1^(4-3)x^3+(4!)/(4!(4-4)!)1^(4-4)x^4`

`1+4x+6x^2+4x^3+x^4`