Maths-
General
Easy

Question

Let , alpha comma beta comma gamma be the roots of (x – a) (x – b) (x – c) = d, , then the roots of the equation left parenthesis straight x minus straight alpha right parenthesis left parenthesis straight x minus straight beta right parenthesis left parenthesis straight x minus straight gamma right parenthesis plus straight d equals 0 are :

  1. a + 1, b + 1, c + 1

  2. a, b, c

  3. a – 1, b – 1, c – 1

  4. a over b comma b over c comma c over a

The correct answer is:

a, b, c


    Step by step solution:
    Given, (x – a) (x – b) (x – c) = d
    rightwards double arrow left parenthesis x squared minus a x minus b x plus a bright parenthesis left parenthesis x minus c right parenthesis equals d
    rightwards double arrow x cubed minus a x squared minus b x squared minus a b xnegative c x squared plus a c x
plus b c x minus a b c equals d
    rightwards double arrow x cubednegative left parenthesis a plus b plus c right parenthesis x squaredplus left parenthesis a b plus b c plus a c right parenthesis xnegative left parenthesis a b c plus d right parenthesis equals 0

    Therefore, we have
    alpha plus beta plus gamma space equals space a plus b plus c
    alpha beta plus beta gamma plus gamma alpha spaceequals space a b plus b c plus c a
    alpha beta gamma equals a b c plus d
    For equation, left parenthesis x minus alpha right parenthesis left parenthesis x minus beta right parenthesisleft parenthesis x minus gamma right parenthesis plus d equals 0
    we have, alpha plus beta plus gamma equals a plus b plus c
    alpha beta plus beta gamma plus gamma alpha spaceequals space a b plus b c plus c a
    and, alpha beta gamma equals a b c plus d
    Thus, a, b, c are the roots of the equation left parenthesis x minus alpha right parenthesis left parenthesis x minus beta right parenthesisleft parenthesis x minus gamma right parenthesis plus d equals 0.
    Hence, option(c) is the correct option.

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