If α,β,1 are the roots of x3−2x2−5x+6=0 then find α,β

1.	If α,β,1 are the roots of x3−2x2−5x+6=0 then find α,β
Answer» Given equation is x³ – 2x² – 5x + 6 = 0 since, 1 is root of the equation, therefore (x + 1) is a factor of it. ⇒ (x – 1)(x² – x – 6) = 0 ⇒ (x – 1)(x² – 3x + 2x – 6) = 0 ⇒ (x – 1)(x(x – 3) + 2(x – 3)) = 0 ⇒ (x – 1)(x + 2)(x – 3) = 0 ⇒ x = 1, –2 or 3. Hence, 1 –2 & 3 are roots of given equation. Then α & β are –2 & 3. i.e, either α = –2 & β = 3 or α = 3 & β = –2.

Answer»

Given equation is

x³ – 2x² – 5x + 6 = 0

since, 1 is root of the equation, therefore (x + 1) is a factor of it.

⇒ (x – 1)(x² – x – 6) = 0

⇒ (x – 1)(x² – 3x + 2x – 6) = 0

⇒ (x – 1)(x(x – 3) + 2(x – 3)) = 0

⇒ (x – 1)(x + 2)(x – 3) = 0

⇒ x = 1, –2 or 3.

Hence, 1 –2 & 3 are roots of given equation.

Then α & β are –2 & 3.

i.e, either α = –2 & β = 3

or α = 3 & β = –2.

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If α,β,1 are the roots of x3−2x2−5x+6=0 then find α,β

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