If ax2 + bx + c and bx2 + ax + c have a common factor x + 1 then show

1.	If ax2 + bx + c and bx2 + ax + c have a common factor x + 1 then show that c = 0 and a = b.
Answer» Let f(x) = ax² + bx + c and g(x) = bx² + ax + c given that (x + 1) is a common factor for both f(x) and g(x). ∴ f(-1) = g(- 1) ⇒a(- 1)² + b(- 1) + c = b(- 1)² + a (- 1) + c ⇒ a – b + c = b – a + c ⇒ a + a = b + b ⇒ 2a = 2b ⇒ a = b Also f(- 1) = a – b + c = 0 ⇒ b – b + c = 0 ⇒ c = 0

If ax2 + bx + c and bx2 + ax + c have a common factor x + 1 then show that c = 0 and a = b.

Answer»

Let f(x) = ax² + bx + c and g(x) = bx² + ax + c

given that (x + 1) is a common factor for both f(x) and g(x).

∴ f(-1) = g(- 1)

⇒a(- 1)² + b(- 1) + c

= b(- 1)² + a (- 1) + c

⇒ a – b + c = b – a + c

⇒ a + a = b + b

⇒ 2a = 2b

⇒ a = b

Also f(- 1) = a – b + c = 0

⇒ b – b + c = 0

⇒ c = 0