Given that the zeros of the cubic polynomial x3 - 6x2 + 3x + 10 are from a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well the aeros of the given polynomial
Answers
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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
May be it's helpful for you
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