Question

How do you use the Binomial Theorem to expand `(x + z) ^ 5`?

Answer 1

`=x^5+5x^4z+10x^3z^2+10x^2z^3+5xz^4+z^5`

Explanation:

The binomial theorem states that

`(x+y)^n+sum_(r=1)^n ""^nC_rx^(n-r)y^r`

`therefore (x+z)^5=sum_(r=1)^5""^5C_rx^(5-r)z^r`

`=""^5C_0x^5+""^5C_1x^(5-1)z^1+""^5C_2x^(5-2)z^2+""^5C_3x^(5-3)z^3+""^5C_4x^(5-4)z^4+""^5C_5x^(5-5)z^5`

`=x^5+5x^4z+10x^3z^2+10x^2z^3+5xz^4+z^5`.