Question

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

Answer 1

Explanation.

⇒ ax²+bx+c and  bx²+ax+c  have common factor ( x + 1 ).

To show that c = 0 and a = b.

⇒ ax²+bx+c + 0.    ..........(1).

common factor is equal to zeroes.

⇒  x + 1 = 0.

⇒ x = -1.

put the value of x = -1 in equation we get,

⇒ a(-1)²+b(-1)+c = 0.

⇒ a - b + c = 0.

⇒ a + c = b   .........(3).

⇒ bx²+ax+c = 0.    ..........(2).

put the value of x = -1 in equation we get,

⇒ b(-1)²+a(-1)+c = 0

⇒ b - a + c = 0.

⇒ b + c = a    ............(4).

put the value of equation (4) in equation (3) we get,

⇒ b + c + c = b.

⇒ 2c = 0.

⇒ c = o.

put the value of c = 0 in equation (3) and (4) we get,

put the value in equation (3).

⇒ a + 0 = b.

⇒ a = b.

put the value in equation (4)

⇒ b + 0 = a.

⇒ b = a.

HENCE VERIFIED.

                                       

MORE INFORMATION.

(1) = D = Discriminant = b²-4ac =0

(2) = value of x = $\frac{-b\pm \sqrt{d} }{2a}$.

(3) = quadratic polynomial whose zeroes are = α  and  β.

       x²-(α+β)x+αβ = 0.

(4) = quadratic polynomial whose zeroes are = α,β and γ.

       x³-(α+β+γ)x²+(αβ+βγ+γα)x-(αβγ) = 0.