Question

Calcule o coeficiente do termo x^2 no desenvolvimento pelo binômino de Newton da expressão(x+2)^6

Answer 1

$Termo \ geral:\\ (a+b)^n\\ T_{p+1}=\binom{n}{p}a^{n-p}.b^p\\\\ T_{p+1}=\binom{6}{p}x^{6-p}.2^p\\\\ T_{p+1}=\binom{6}{p}x^{6-p}.2^p\\\\ Como \ queremos \ x^2\\ x^{6-p}=x^2\\ 6-p=2\\ p=6-2\\ p=4\\ Assim:\\ T_{4+1}=\binom{6}{4}x^{6-4}.2^4\\\\ T_{5}=\binom{6}{4}x^{2}.16\\\\ T_{5}=\frac{6!}{4!2!}.x^{2}.16\\\\ T_{5}=\frac{6.5}{2.1}.x^{2}.16\\\\ T_{5}=\frac{30}{2}.x^{2}.16\\\\ T_{5}=15.x^{2}.16\\\\ T_{5}=240.x^{2}\\ O \ coeficiente \ \'e \ 240.$