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If 3x^3 + ax^2 + 5x = 3(x+b)^3 + c, find all possible values of a, b, and c.

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  • 6 months ago

    Hint: expand the right hand side & compare

    the coefficients.

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  • 6 months ago

    3x^3 + ax^2 + 5x = 3(x + b)^3 + c

    a x^2 + 3 x^3 + 5 x = 3 b^3 + 9 b^2 x + 9 b x^2 + c + 3 x^3

    a = -3 sqrt(5), b = -sqrt(5)/3, c = (5 sqrt(5))/9

    a = 3 sqrt(5), b = sqrt(5)/3, c = -(5 sqrt(5))/9

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  • 6 months ago

    Expand the right side:

    3x^3 + ax^2 + 5x = 3(x^3 = 3bx^2 + 3b^2x + b^3) + c

    (3x^3 + ax^2 + 5x = (3)x^3 + (9b)x^2 + (9b^2)x + (3b^3 + c)

    Equate coefficients on like powers of x:

    3 = 3

    a = 9b

    5 = 9b^2

    0 = 3b^3 + c

    The the third line says that b must be ±(√5)/3. Then line 2 says

    a = 9b = ±3√5 . . . . same sign as b

    c = -3b^3 = - ± 3(5√5) / (3^3) = - ±5√5 / 9. . . . opposite sign of b

    So (a, b, c) = (3√5.√5 / 3, -5√5 / 9) or (-3√5, -√5 / 3, 5√5 / 9).

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