desarrollar (z^2+4x)^4

Solución

desarrollar

(z^{2} + 4 x)^{4}

z^{8} + 16 z^{6} x + 96 z^{4} x^{2} + 256 z^{2} x^{3} + 256 x^{4}

Pasos de solución

(z^{2} + 4 x)^{4}

Aplicar el teorema del binomio:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = z^{2}, b = 4 x

= i = 0 \sum 4 (i 4) (z^{2})^{(4 - i)} (4 x)^{i}

Expandir sumatorio

= \frac{4 !}{0 ! ( 4 - 0 ) !} (z^{2})^{4} (4 x)^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} (z^{2})^{3} (4 x)^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} (z^{2})^{2} (4 x)^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} (z^{2})^{1} (4 x)^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} (z^{2})^{0} (4 x)^{4}

Simplificar

\frac{4 !}{0 ! ( 4 - 0 ) !} (z^{2})^{4} (4 x)^{0} : z^{8}

Simplificar

\frac{4 !}{1 ! ( 4 - 1 ) !} (z^{2})^{3} (4 x)^{1} : 16 z^{6} x

Simplificar

\frac{4 !}{2 ! ( 4 - 2 ) !} (z^{2})^{2} (4 x)^{2} : 96 z^{4} x^{2}

Simplificar

\frac{4 !}{3 ! ( 4 - 3 ) !} (z^{2})^{1} (4 x)^{3} : 256 z^{2} x^{3}

Simplificar

\frac{4 !}{4 ! ( 4 - 4 ) !} (z^{2})^{0} (4 x)^{4} : 256 x^{4}

= z^{8} + 16 z^{6} x + 96 z^{4} x^{2} + 256 z^{2} x^{3} + 256 x^{4}

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