1.

if the zeroes of the cubic polynomial x(cube) - 6x(square) +3x +10 are of the form a, a+b, a+2b for some real number a and b, find the value of a and b as well as the zeroes of the given polynomial.

Answer»

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


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