Question

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

Answer 1

Answer:

Given

ax^2 + bx + c and bx^2 + ax + c have common factor ( x +1 )

so,

x+1=0

x= -1

p(x) = ax^2 + bx + c = 0

p(-1) = a(-1)^2 + b(-1) + c = 0

= a + (-b) +c = 0

= a-b+c = 0 ..........(i)

now

p(x) = bx^2 + ax + c = 0

p(-1) = b(-1)^2 + a(-1) + c =0

= b + (-a) +c = 0

= b-a+c = 0 ...........(ii)

now adding equation (i) and (ii)

a-b+c+b-a+c = 0

2c = 0

c= 0 ( proved)

now

from equation (I) and (ii)

a-b+c = b-a+c

a+a = b+b

2a = 2b

hence a= b

hope this will help you