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受欢迎的代数 >

展开 (5+x/2)^4

解答

展开

(5 + \frac{x}{2})^{4}

解答

625 + 250 x + \frac{75 x ^{2}}{2} + \frac{5 x ^{3}}{2} + \frac{x ^{4}}{16}

求解步骤

(5 + \frac{x}{2})^{4}

使用二项式定理:

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

a = 5, b = \frac{x}{2}

= i = 0 \sum 4 (i 4) \cdot 5^{(4 - i)} (\frac{x}{2})^{i}

展开求和

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

化简

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 5^{4} (\frac{x}{2})^{0} : 625

化简

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 5^{3} (\frac{x}{2})^{1} : 250 x

化简

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 5^{2} (\frac{x}{2})^{2} : \frac{75 x ^{2}}{2}

化简

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 5^{1} (\frac{x}{2})^{3} : \frac{5 x ^{3}}{2}

化简

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 5^{0} (\frac{x}{2})^{4} : \frac{x ^{4}}{16}

= 625 + 250 x + \frac{75 x ^{2}}{2} + \frac{5 x ^{3}}{2} + \frac{x ^{4}}{16}

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