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Popular Álgebra >

desarrollar (x-pi)^7

Solución

desarrollar

(x - π)^{7}

Solución

x^{7} - 7 π x^{6} + 21 π^{2} x^{5} - 35 π^{3} x^{4} + 35 π^{4} x^{3} - 21 π^{5} x^{2} + 7 π^{6} x - π^{7}

Pasos de solución

(x - π)^{7}

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 7 (i 7) x^{(7 - i)} (- π)^{i}

Expandir sumatorio

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

Simplificar

\frac{7 !}{0 ! ( 7 - 0 ) !} x^{7} (- π)^{0} : x^{7}

Simplificar

\frac{7 !}{1 ! ( 7 - 1 ) !} x^{6} (- π)^{1} : - 7 π x^{6}

Simplificar

\frac{7 !}{2 ! ( 7 - 2 ) !} x^{5} (- π)^{2} : 21 π^{2} x^{5}

Simplificar

\frac{7 !}{3 ! ( 7 - 3 ) !} x^{4} (- π)^{3} : - 35 π^{3} x^{4}

Simplificar

\frac{7 !}{4 ! ( 7 - 4 ) !} x^{3} (- π)^{4} : 35 π^{4} x^{3}

Simplificar

\frac{7 !}{5 ! ( 7 - 5 ) !} x^{2} (- π)^{5} : - 21 π^{5} x^{2}

Simplificar

\frac{7 !}{6 ! ( 7 - 6 ) !} x^{1} (- π)^{6} : 7 π^{6} x

\frac{7 !}{7 ! ( 7 - 7 ) !} x^{0} (- π)^{7} = - π^{7}

= x^{7} - 7 π x^{6} + 21 π^{2} x^{5} - 35 π^{3} x^{4} + 35 π^{4} x^{3} - 21 π^{5} x^{2} + 7 π^{6} x - π^{7}

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