Question

If a not equal to b and x²+ax+b and x²+bx+a have a common factor, show that a+b+1=0

Answer 1

If P(x) = x2 + ax + b
and
Q(x) = x2 + bx + a

have a common factor say (x - β), then

P(β) = 0 and Q(β) = 0

⇒ β2 + aβ + b = 0 ... (1)

and β2 + bβ + a = 0 ... (2)

On subtracting (1) from (2), we get

(a-b)β + (b-a) = 0

⇒ (a-b)β - (a-b) = 0

⇒ (a-b)β = (a-b)

⇒ β = 1

On putting β = 1 in (1), we get

12 + a.1 + b = 0

⇒ a + b = -1

so it is proved a+b+1=0