Question

if alpha be a multiple root of the order 3 of the equation x^4+bx^2+cx+d=0 then prove alpha= 8d/3c​

Answer 1

Step-by-step explanation:

$.$

hence, the roots can be assumed as α,α,α,β

Here, S

1

=3α+β=−a;S

2

=3α(α+β)=b;S

3

=α

2

(α+3β)=−c;S

4

=α

3

β=d

We need to evaluate the value of

3a

2

−8b

6c−ab

6c−ab=α(3α

2

−6αβ+3β

2

)

3a

2

−8b=3α

2

−6αβ+3β

2

∴

3a

2

−8b

6c−ab

=

3α

2

−6αβ+3β

2

α(3α

2

−6αβ+3β

2

)

=α, which is the common root of the given equation

Answer 2

Answer:

Let other root be y,

Now roots are alpha(3), y.

sum of roots = 3alpha+y = 0

product of roots = 3alpha.y = -d

Now, solve these equations. You get the proof required.