4) 8 -B = UIf a and b are unequal and x² + ax + b and x+ bx + a have

1.	4) 8 -B = UIf a and b are unequal and x² + ax + b and x+ bx + a have a common factor, then
Answer» If P(x) = x^2+ ax + b and Q(x) = x^2+ bx + a have a common factor say (x - β), then P(β) = 0 and Q(β) = 0 ⇒ β^2+ aβ + b = 0 ... (1) and β^2+ 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

4) 8 -B = UIf a and b are unequal and x² + ax + b and x+ bx + a have a common factor, then

Answer»

If P(x) = x^2+ ax + b and Q(x) = x^2+ bx + a have a common factor say (x - β), then

P(β) = 0 and Q(β) = 0

⇒ β^2+ aβ + b = 0 ... (1)

and β^2+ 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