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受欢迎的代数 >

展开 (3x+5y)^4

解答

展开

(3 x + 5 y)^{4}

解答

81 x^{4} + 540 x^{3} y + 1350 x^{2} y^{2} + 1500 x y^{3} + 625 y^{4}

求解步骤

(3 x + 5 y)^{4}

使用二项式定理:

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

a = 3 x, b = 5 y

= i = 0 \sum 4 (i 4) (3 x)^{(4 - i)} (5 y)^{i}

展开求和

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

化简

\frac{4 !}{0 ! ( 4 - 0 ) !} (3 x)^{4} (5 y)^{0} : 81 x^{4}

化简

\frac{4 !}{1 ! ( 4 - 1 ) !} (3 x)^{3} (5 y)^{1} : 540 x^{3} y

化简

\frac{4 !}{2 ! ( 4 - 2 ) !} (3 x)^{2} (5 y)^{2} : 1350 x^{2} y^{2}

化简

\frac{4 !}{3 ! ( 4 - 3 ) !} (3 x)^{1} (5 y)^{3} : 1500 x y^{3}

化简

\frac{4 !}{4 ! ( 4 - 4 ) !} (3 x)^{0} (5 y)^{4} : 625 y^{4}

= 81 x^{4} + 540 x^{3} y + 1350 x^{2} y^{2} + 1500 x y^{3} + 625 y^{4}

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