desarrollar (x-pi)^8

Solución

desarrollar

(x - π)^{8}

x^{8} - 8 π x^{7} + 28 π^{2} x^{6} - 56 π^{3} x^{5} + 70 π^{4} x^{4} - 56 π^{5} x^{3} + 28 π^{6} x^{2} - 8 π^{7} x + π^{8}

Pasos de solución

(x - π)^{8}

Aplicar el teorema del binomio:

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

a = x, b = - π

= i = 0 \sum 8 (i 8) x^{(8 - i)} (- π)^{i}

Expandir sumatorio

= \frac{8 !}{0 ! ( 8 - 0 ) !} x^{8} (- π)^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} x^{7} (- π)^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} x^{6} (- π)^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} x^{5} (- π)^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} x^{4} (- π)^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} x^{3} (- π)^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} x^{2} (- π)^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} x^{1} (- π)^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} x^{0} (- π)^{8}

Simplificar

\frac{8 !}{0 ! ( 8 - 0 ) !} x^{8} (- π)^{0} : x^{8}

Simplificar

\frac{8 !}{1 ! ( 8 - 1 ) !} x^{7} (- π)^{1} : - 8 π x^{7}

Simplificar

\frac{8 !}{2 ! ( 8 - 2 ) !} x^{6} (- π)^{2} : 28 π^{2} x^{6}

Simplificar

\frac{8 !}{3 ! ( 8 - 3 ) !} x^{5} (- π)^{3} : - 56 π^{3} x^{5}

Simplificar

\frac{8 !}{4 ! ( 8 - 4 ) !} x^{4} (- π)^{4} : 70 π^{4} x^{4}

Simplificar

\frac{8 !}{5 ! ( 8 - 5 ) !} x^{3} (- π)^{5} : - 56 π^{5} x^{3}

Simplificar

\frac{8 !}{6 ! ( 8 - 6 ) !} x^{2} (- π)^{6} : 28 π^{6} x^{2}

Simplificar

\frac{8 !}{7 ! ( 8 - 7 ) !} x^{1} (- π)^{7} : - 8 π^{7} x

\frac{8 !}{8 ! ( 8 - 8 ) !} x^{0} (- π)^{8} = π^{8}

= x^{8} - 8 π x^{7} + 28 π^{2} x^{6} - 56 π^{3} x^{5} + 70 π^{4} x^{4} - 56 π^{5} x^{3} + 28 π^{6} x^{2} - 8 π^{7} x + π^{8}

e x p an d (x - π)^{6}

e x p an d \frac{x ( x - 1 )}{20}

e x p an d \frac{( 9 x - 2 ) ^{3}}{7}

e x p an d \frac{1}{v ( m - 3 )}

e x p an d - 2 (2 z + 1 = 26)