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1. given that the zeroes of the cubic polynimial x 3 - 6x 2 +3x+10 are of the form a, a+b, a+2b for some real no.s a and b. find the values of a and b as well as the zeroes of the given polynomial

1. given that the zeroes of the cubic polynimial   x3 - 6x2+3x+10 are of the form a, a+b, a+2b for some real no.s a and b. find the values of a and b as well as the zeroes of the given polynomial

    Grade:10

    1 Answers

    Arun
    25763 Points
    3 years ago
    Dear student
     
    Given that a, a+b, a+2b are roots of given polynomial x³-6x²+3x+10

    From this polynomial,
    Sum of the roots     ⇒ a+2b+a+a+b = -coefficient of x²/ coefficient of x³
                               ⇒ 3a+3b = -(-6)/1 = 6
                               ⇒ 3(a+b) = 6
                               ⇒ a+b = 2  --------- (1)  b = 2-a

    Product of roots ⇒ (a+2b)(a+b)a = -constant/coefficient of x³
                              ⇒ (a+b+b)(a+b)a = -10/1

    Placing the value of a+b=2 in it
                             ⇒ (2+b)(2)a = -10
                             ⇒ (2+b)2a = -10
                             ⇒ (2+2-a)2a = -10
                             ⇒ (4-a)2a = -10
                             ⇒ 4a-a² = -5
                             ⇒ a²-4a-5 = 0
                             ⇒ a²-5a+a-5 = 0
                             ⇒ (a-5)(a+1) = 0
                          a-5 = 0      or        a+1 = 0
                          a = 5                     a = -1                

      a = 5, -1 in (1) a+b = 2

    When a = 5, 5+b=2 ⇒ b=-3
               a = -1, -1+b=2 ⇒ b= 3

    ∴ If a=5 then b= -3
                or
       If a= -1 then b=3
     
    Regards
    Arun (askIITians forum expert)

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