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

x-1 is the factor of two polynomials means x=1 satisfies two equation​
a-b+c=0
b-a+c=0
so,
a-b+c=b-a+c
a-b=0
putting this 1st equation
we get
c=0

Answer 2

x-1=0
x=1
p(x)=ax^2-bx+c
f(x)=bx^2-ax+c
As they have a common factor they both leave a remainder zero when divided by x-1
f(x)=0
p(x)=0
a(1)^2-b(1)+c=0
a-b+c=0.........(1)

b(1)^2-a(1)+c=0
b-a+c=0......(2)
(1)+(2)
a-b+c=0
-a+b+c=0
a and b get cancelled
2c=0
c=0

Substituting c in eq 1
a-b+0=0
a=b
Hence proved


Hope it helps!!!