How do you use the binomial series to expand ${(5 + x)}^{4}$ $(5 + x)^4$ ?

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Answer 1 · 2018-04-17T13:15:53

${(5 + x)}^{4} = 625 + 500 x + 150 x^{2} + 20 x^{3} + x^{4}$ $(5+x)^4=625+500x+150x^2+20x^3+x^4$

The binomial series expansion for $(a+bx)^n,ninZZ;n>0$ is given by:
$(a+bx)^n=sum_(r=0)^n((n!)/(r!(n-1)!)a^(n-r)(bx)^r)$

So, we have:
$(5+x)^4=(4!)/(0!*4!)5^4+(4!)/(1!*3!)(5)^3x+(4!)/(2!*2!)(5)^2x^2+(4!)/(4!*1!)(5)x^3+(4!)/(4!*0!)x^4$

$(5+x)^4=5^4+4(5)^3x+6(5)^2x^2+4(5)x^3+x^4$

$(5+x)^4=625+500x+150x^2+20x^3+x^4$

How do you use the binomial series to expand (5 + x)^4(5+x)4(5 + x)^4?