Question

If p+iq be one of the roots of equation x^3+ax+b=0 then 2p is one of the roots if equation

Answer 1

Answer with explanation:

It is given that, p+i q , is one of the roots of the equation

      x³+ a x +b=0

This is a cubic equation.So, it has three roots.

As,Imaginary root occur in pairs.

So, if , p + i q,is one root of the equation , then , p - i q, will be another root.It is given that third Root is equal to 2 p.

For, any Cubic equation of the form

  $Ax^3+Bx^2+Cx+D=0$

If, u,v and w are roots of the equation

   $u+v+w=\frac{-B}{A}\\\\u v+v w+w u=\frac{C}{A}\\\\uvw=\frac{-D}{A}$

Then, ⇒⇒p+i q + p- i q + 2 p=0

        4 p=0

p=0

or

Also,⇒ (p +i q)(p-i q)×2 p= -b

⇒(p²+q²)×2 p= -b

⇒2 p³+2 p q² + b=0

or

⇒(p+ i q)× (p -i q) + 2 p× ( p + i q)+2 p×(p- i q)=a

⇒p²+q²+2 p²+2 p i q+2 p² -2 p i q=a

⇒ 5 p² +q²-a=0