Question

Expand the following binomial:
${(x^2+x)}^5$

Answer 1

using binomial theorem

${(x+y)}^n$

$=\sum _{r=0}^n{}_r^{n}C{x}^{n-r}{y}^r$

for the given expression

$(x^2+x{)}^5$

$={}_0^5{C{\left(x^2\right)}^5{\left(x\right)}^0+}_1^5{C{\left(x^2\right)}^4{\left(x\right)}^1+}_2^{5}C{\left(x^2\right)}^3{\left(x\right)}^2{+}_3^5{C{\left(x^2\right)}^2{\left(x\right)}^3+}_4^{5}C{\left(x^2\right)}^1{\left(x\right)}^4{+}_5^{5}C{\left(x^2\right)}^0{\left(x\right)}^5$

$=x^{10}+5x^9+10x^8+10x^7+5x^6+x^5$