Question

Binômio de Newton: (3a-4)⁴

Answer 1

Olá Cristy,

dado o binômio,

$(3a-4)^4$

podemos usar a fórmula binomial,

$T_{p+1} =\left(\begin{array}{ccc}n\\p\end{array}\right)*a^{n-p}*b~^p$

e desenvolvê-lo, acompanhe:

$(3a-4)^4= \left(\begin{array}{ccc}4\\0\end{array}\right)*3a^{4-0}*(-4)^0\\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\to~\dfrac{4!}{0!(4-0)!}= \dfrac{4!}{1!4!}=1*3a^4*1 \\\\ (3a-4)^4=\left(\begin{array}{ccc}4\\1\end{array}\right)*3a^{4-1}*(-4)^1\\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\to~\dfrac{4!}{1!(4-1)!}= \dfrac{4!}{1!3!}=4*3a^3*(-4)$

$(3a-4)^4= \left(\begin{array}{ccc}4\\2\end{array}\right)*3a^{4-2} *(-4)^2\\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \dfrac{4!}{2!(4-2)!}= \dfrac{4!}{2!2!}=6*3a^2*16\\\\\\ (3a-4)^4= \left(\begin{array}{ccc}4\\3\end{array}\right)*3a^{4-3}*(-4)^3\\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \dfrac{4!}{3!(4-3)!}= \dfrac{4!}{3!1!}=6*3a^1*(-64)$

$(3a-4)^4= \left(\begin{array}{ccc}4\\4\end{array}\right)*3a^{4-4}*(-4)^4\\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \dfrac{4!}{4!(4-4)!}= \dfrac{4!}{4!0!}=1*3a^0*256\\\\\\ (3a-4)^4=1*3a^4*1+4*3a^3*(-4)+6*3a^2*16+6*3a^1*(-64)\\ +1*3a^0*256\\\\O~desenvolvimento~do~binomio~sera:\\\\ \boxed{(3a-4)^4=3a^4-48a^3+288a^2-1.152a+256}$

Espero ter ajudado e tenha ótimos estudos =))