求证(a-b)^3=a^3-3a^2b+3ab^2-b^3用数学符号或概念来答

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求证(a-b)^3=a^3-3a^2b+3ab^2-b^3用数学符号或概念来答

求证(a-b)^3=a^3-3a^2b+3ab^2-b^3用数学符号或概念来答
求证(a-b)^3=a^3-3a^2b+3ab^2-b^3
用数学符号或概念来答

求证(a-b)^3=a^3-3a^2b+3ab^2-b^3用数学符号或概念来答
证明:
∵(a-b)³
=(a-b)(a-b)²
=(a-b) (a²-2ab+b²)
=a(a²-2ab+b²)-b(a²-2ab+b²)
=a³-2a²b+ab²-a²b+2ab²-b³
=a³-3a²b+3ab²-b³
∴(a-b)³=a³-3a²b+3ab²-b³

(a-b)^3=a^3-3a^2b+3ab^2-b^3
a^3-b^3=(a-b)^3-[-3(a^2)b+3ab^2]
=(a-b)(a-b)^2+3ab(a-b)
=(a-b)(a^2-2ab+b^2+3ab)
=(a-b)(a^2+ab+b^2)
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左边=(a-b)^3
=(a-b)²(a-b)
=(a²-2ab+b²)(a-b)
=a³-2a²b+ab²-a²b+2ab²-b³
=a^3-3a^2b+3ab^2-b^3=右边
得证。

a^3-3a^2b+3ab^2-b^3
=a^3-b^3-3a^2b+3ab^2
=(a-b)(a^2+ab+b^2)-3ab(a-b)
=(a-b)(a^2+ab+b^2-3ab)
=(a-b)(a-b)^2
=(a-b)^3
左边=右边

(a-b)^3=a^3-3a^2b+3ab^2-b^3