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

1 Answer
Feb 25, 2016

$= {x}^{5} + 5 {x}^{4} z + 10 {x}^{3} {z}^{2} + 10 {x}^{2} {z}^{3} + 5 x {z}^{4} + {z}^{5}$

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

$= {\text{^5C_0x^5+""^5C_1x^(5-1)z^1+""^5C_2x^(5-2)z^2+""^5C_3x^(5-3)z^3+""^5C_4x^(5-4)z^4+}}^{5} {C}_{5} {x}^{5 - 5} {z}^{5}$

$= {x}^{5} + 5 {x}^{4} z + 10 {x}^{3} {z}^{2} + 10 {x}^{2} {z}^{3} + 5 x {z}^{4} + {z}^{5}$.