Question

5- Calcule o seguinte binomio
(x-4) elevado a 6

Answer 1

Explicação passo-a-passo:

De modo geral:

$\sf (a+b)^n=\dbinom{n}{0}\cdot a^n\cdot b^0+\dbinom{n}{1}\cdot a^{n-1}\cdot b^1+\dots+\dbinom{n}{n-1}\cdot a^1\cdot b^{n-1}+\dbinom{n}{n}\cdot a^0\cdot b^n$

Assim:

$\sf (x-4)^6=\dbinom{6}{0}\cdot x^6\cdot(-4)^0+\dbinom{6}{1}\cdot x^5\cdot(-4)^1+\dbinom{6}{2}\cdot x^4\cdot(-4)^2+\dbinom{6}{3}\cdot x^3\cdot(-4)^3+\dbinom{6}{4}\cdot x^2\cdot(-4)^4+\dbinom{6}{5}\cdot x^1\cdot(-4)^5+\dbinom{6}{6}\cdot x^0\cdot(-4)^6$

• Os números binomiais valem:

$\sf \dbinom{6}{0}=1$

$\sf \dbinom{6}{1}=6$

$\sf \dbinom{6}{2}=\dfrac{6\cdot5}{2!}=\dfrac{30}{2}=15$

$\sf \dbinom{6}{3}=\dfrac{6\cdot5\cdot4}{3!}=\dfrac{120}{6}=20$

$\sf \dbinom{6}{4}=\dbinom{6}{2}=\dfrac{6\cdot5}{2!}=\dfrac{30}{2}=15$

$\sf \dbinom{6}{5}=\dbinom{6}{1}=6$

$\sf \dbinom{6}{6}=\dbinom{6}{0}=1$

Logo:

$\sf (x-4)^6=\dbinom{6}{0}\cdot x^6\cdot(-4)^0+\dbinom{6}{1}\cdot x^5\cdot(-4)^1+\dbinom{6}{2}\cdot x^4\cdot(-4)^2+\dbinom{6}{3}\cdot x^3\cdot(-4)^3+\dbinom{6}{4}\cdot x^2\cdot(-4)^4+\dbinom{6}{5}\cdot x^1\cdot(-4)^5+\dbinom{6}{6}\cdot x^0\cdot(-4)^6$

$\sf (x-4)^6=1\cdot x^6\cdot1+6x^5\cdot(-4)+15x^4\cdot16+20x^3\cdot(-64)+15x^2\cdot256+6x\cdot(-1024)+1\cdot4096$

$\sf \red{(x-4)^6=x^6-24x^5+240x^4-1280x^3+3840x^2-6144x+4096}$