polynomials (1 Viewer)

[maths lover]

hey guys i am just having troubles with a polynomials question. ok here goes
if one root of the equation x^3-bx^2+cx-d=0 is equal to the product of the other two, show that (c+d)^2=d(b+1)^2
thanks in advance.

 

[bleakarcher]

x^3-bx^2+cx-d=0
Let the roots be A, B, AB.
A+B+AB=b...(1)
AB+BA^2+AB^2=c...(2)
A^2*B^2=(AB)^2=d...(3)
Rewriting (1),
AB(1+A+B)=c...(4)
Rewriting (3),
AB=d^(1/2)...(5)
Sub AB in (1),
A+B=b-d^(1/2)...(6)
Sub AB, (A+B) in (4),
d^(1/2)[1+b-d^(1/2)]=c
d^(1/2)+bd^(1/2)-d=c
(c+d)^2=[d^(1/2)+bd^(1/2)]^2
(c+d)^2=[d^(1/2)[1+b]]^2
(c+d)^2=d(1+b)^2 as req.