Question

Find α and β if x+1 and x+2 are factors of x³+3x²-2αx+β.

Answer 1

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Answer 2

Let, f (x) = x³ + 3x² - 2αx + β  be the given polynomial.

From factor theorem, If (x + 1) and (x + 2) are factors of f (x) then f (-1) = 0 and f (-2) = 0 :  

f (-1) = 0

⇒(-1)³ + 3 (-1)² - 2 α (-1) + β = 0

⇒ - 1 + 3 + 2 α + β = 0

⇒ 2 α + β + 2 = 0 ……………(1)

Similarly,

f (- 2) = 0

⇒ (-2)³ + 3 (- 2)² – 2α (- 2) + β = 0

⇒ - 8 + 12 + 4 α + β = 0

⇒ 4 α + β + 4 = 0 ……………(2)

Subtract eq (1) from eq (2) :  

⇒ 4 α + β + 4 – (2 α + β + 2) = 0 - 0

⇒ 4 α + β + 4 - 2 α - β - 2 = 0

⇒ 4 α - 2 α + β  - β + 4 - 2 = 0

⇒ 2 α + 2 = 0

⇒ 2 α = 0 - 2

⇒ 2 α = - 2

⇒ α = - 2/2

⇒ α = -1

On Putting α = -1 in eq (1) :  

⇒ 2 α + β + 2 = 0

⇒ 2 (-1) + β + 2 = 0

⇒ - 2  + β + 2 = 0

⇒ β = 0

Hence, α = -1 and β = 0.

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