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Grade: 10

                        

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

2 years ago

Answers : (1)

Arun
24742 Points
							
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)
2 years ago
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