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

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

As (x + 1) is the common factor of f(x) and p(x) both, and as by Factor Theorem, we know that,

If p(x) is a polynomial and a is any real number, then g(x) = (x– a) is a factor of p(x), if p(a) = 0 and vice versa.

⇒ f(–1) = p (–1) = 0

⇒ a(–1)² + b(–1) + c = b(–1)² + a(–1) + c

⇒ a – b + c = b – a + c

⇒ 2a = 2b

⇒ a = b ------ (A)

Also, we discussed that,

f(–1) = 0

⇒ a(–1)² + b(–1) + c = 0

⇒ a – b + c = 0

From equation (A), we see that a = b,

⇒ c = 0 -------- (B)

∴ Equations (A) and (B) show us the required result.