Question

Calculo Desenvolva os Seguintes binomios (x+3)4

Answer 1

Hola! 
Calculo Desenvolva os Seguintes binomios (x+3)^4

$(x+3)^4= (x+3)^2*(x+3)^2 \\ \\ (x+3)^4= (x^2+6x+9)*(x^2+6x+9)^2 \\ \\ (x+3)^4= x^4+6x^3+9x^2+6x^3+36x^2+54x+9x^2+54x+81 \\ \\ (x+3)^4= \boxed{ x^4+12x^3+54x^2+108x+81 }$

Answer 2

Olá!

Temos:

$\left(x+3\right)^4$

Aplicando à fórmula do Binômio de Newton:

$\left(x+a\right)^n=\sum _{p=0}^n\binom{n}{p}x^{\left(n-p\right)}a^p$

dados:

x = x

a = 3

n = 4

logo:

$\left(x+3\right)^4 \to \sum _{p=0}^4\binom{4}{p}x^{\left(4-p\right)}*3^p$

Se:

$\binom{n}{p}=\dfrac{n!}{p!\left(n-p\right)!}$

então, para p = 0, p = 1, p = 2, p = 3, p =4, teremos:

* p = 0

$\dfrac{n!}{p!\left(n-p\right)!}*x^{\left(n-p\right)}*3^p \to \dfrac{4!}{0!\left(4-0\right)!}*x^{\left(4-0\right)}*3^0 \to\dfrac{4!}{0!4!}*x^4*3^0$

$\to\dfrac{\diagup\!\!\!\!4!}{\diagup\!\!\!\!4!}*x^4*1 = \boxed{x^4}$

* p = 1

$\dfrac{n!}{p!\left(n-p\right)!}*x^{\left(n-p\right)}*3^p \to \dfrac{4!}{1!\left(4-1\right)!}*x^{\left(4-1\right)}*3^1 \to\dfrac{4!}{1!3!}*x^3*3^1$

$\to\dfrac{4*\diagup\!\!\!\!3!}{\diagup\!\!\!\!3!}*3x^3 \to 4*3x^3 = \boxed{12x^3}$

* p = 2

$\dfrac{n!}{p!\left(n-p\right)!}*x^{\left(n-p\right)}*3^p \to \dfrac{4!}{2!\left(4-2\right)!}*x^{\left(4-2\right)}*3^2 \to\dfrac{4!}{2!2!}*x^2*9$

$\to\dfrac{\diagup\!\!\!\!4^2*3*\diagup\!\!\!\!2!}{\diagup\!\!\!\!2^1*1*\diagup\!\!\!\!2!}*9x^2 \to 2*3*9x^2 = \boxed{54x^2}$

* p = 3

$\dfrac{n!}{p!\left(n-p\right)!}*x^{\left(n-p\right)}*3^p \to \dfrac{4!}{3!\left(4-3\right)!}*x^{\left(4-3\right)}*3^3 \to\dfrac{4!}{3!1!}*x^1*27$

$\to\dfrac{4*\diagup\!\!\!\!3!}{\diagup\!\!\!\!3!}*27x \to 4*27x = \boxed{108x}$

* p = 4

$\dfrac{n!}{p!\left(n-p\right)!}*x^{\left(n-p\right)}*3^p \to \dfrac{4!}{4!\left(4-4\right)!}*x^{\left(4-4\right)}*3^4 \to\dfrac{4!}{4!0!}*x^0*81$

$\to\dfrac{\diagup\!\!\!\!4!}{\diagup\!\!\!\!4!}*x^0*81 = \boxed{81}$

Agora, somamos todos os fatores encontrados e assim teremos a resposta final, vejamos:

$\boxed{\boxed{x^4 + 12x^3 + 54x^2 + 108x + 81}}\Longleftarrow(resposta)\end{array}}\qquad\checkmark$

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Espero ter ajudado, saudações, DexteR! =)