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受欢迎的代数 >

展开 (4x-pi)^5

解答

展开

(4 x - π)^{5}

解答

1024 x^{5} - 1280 π x^{4} + 640 π^{2} x^{3} - 160 π^{3} x^{2} + 20 π^{4} x - π^{5}

求解步骤

(4 x - π)^{5}

使用二项式定理:

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

a = 4 x, b = - π

= i = 0 \sum 5 (i 5) (4 x)^{(5 - i)} (- π)^{i}

展开求和

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

化简

\frac{5 !}{0 ! ( 5 - 0 ) !} (4 x)^{5} (- π)^{0} : 1024 x^{5}

化简

\frac{5 !}{1 ! ( 5 - 1 ) !} (4 x)^{4} (- π)^{1} : - 1280 π x^{4}

化简

\frac{5 !}{2 ! ( 5 - 2 ) !} (4 x)^{3} (- π)^{2} : 640 π^{2} x^{3}

化简

\frac{5 !}{3 ! ( 5 - 3 ) !} (4 x)^{2} (- π)^{3} : - 160 π^{3} x^{2}

化简

\frac{5 !}{4 ! ( 5 - 4 ) !} (4 x)^{1} (- π)^{4} : 20 π^{4} x

\frac{5 !}{5 ! ( 5 - 5 ) !} (4 x)^{0} (- π)^{5} = - π^{5}

= 1024 x^{5} - 1280 π x^{4} + 640 π^{2} x^{3} - 160 π^{3} x^{2} + 20 π^{4} x - π^{5}

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