How do you use the binomial theorem to expand and simplify the expression #(x^2+5)^4#?

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Answer 1 · 2017-02-09T10:23:05

The answer is #=x^8+20x^6+150x^4+500x^2+625#

#(a+b)^n=((n),(0))a^nb^0+((n),(1))a^(n-1)b+((n),(2))a^(n-2)b^2+......((n),(n))a^0b^n#

Where,

#((n),(k))=(n!)/((n-k)!k!#

#((4),(0))=(4!)/((4-0)!0! ) =1#

#((4),(1))=(4!)/((4-1)!1!) =4#

#((4),(2))=(4!)/((4-2)!2!)=6#

#((4),(3))=(4!)/((4-3)!3!)=4#

#((4),(4))=(4!)/((4-4)!4!)=1#

#0!=1#

Therefore,

#(x^2+5)^4=(x^2)^4+4(x^2)^3*5+6(x^2)^2*5^2+4(x^2)*5^3+5^4#

#=x^8+20x^6+150x^4+500x^2+625#