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S. a) Given that the zeroes of the cubic polynomial x 1-6x2+3x+10 are of the form a, a+b,a+2b for some real numbers a and b. Find the value of a and b as well as the zeroes ofthe given polynomial. |
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Answer» Given that a, a+b, a+2b are roots of given polynomialx³-6x²+3x+10 From this polynomial,Sum of the roots⇒ a+2b+a+a+b = -coefficient of x²/ coefficient ofx³ ⇒ 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 |
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