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If one root of Ax^(3)+Bx^(2)+Cx+D=0,Dne0...

If one root of `Ax^(3)+Bx^(2)+Cx+D=0,Dne0` is the arithmetic mean of the other two roots, then the relation `2B^(3)+lambdaABC+muA^(2)D=0` holds good. Then, the value of `2lambda+mu` is

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To solve the problem, we need to analyze the given cubic equation and the condition that one root is the arithmetic mean of the other two roots. Let's denote the roots of the cubic equation \( Ax^3 + Bx^2 + Cx + D = 0 \) as \( r_1, r_2, r_3 \). ### Step 1: Set up the roots Since one root is the arithmetic mean of the other two, we can denote the roots as: - \( r_1 = \alpha - d \) - \( r_2 = \alpha \) - \( r_3 = \alpha + d \) ...
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