Question

If the sum of two roots of x^5+ax+b is zero then the value of b is?

Answer 1

Answer:

Step-by-step explanation:

Let  α  and  −α  be the roots as suggested.

plugging  x=α  gives

α5+aα+b=0…..(6)  

plugging  x=−α  gives

−α5−aα+b=0…..(7)  

Adding  (6)  and  (7)   2b=0.  Hence  b=0  

In fact we could state anything like

If the sum of two roots of  x9+ax7+bx5+cx3+dx+e  is zero then the value of  e  is?

……………………………………………………………………………..

Let  p,q,r,s  and  t  be the  5  roots of the equation.

The coefficients of  x4,x3  and  x2  are all  0  in the equation.

Using Vieta’s formula

p+q+r+s+t=0……(1)  

p(q+r+s+t)+q(r+s+t)+r(s+t)+st=0…..(2)  

pq(r+s+t)+pr(s+t)+pst+qr(s+t)+qst+rst=0…..(3)  

In addition it is given sum of  2  roots is  0  .Let  s+t=0….(4)  

plugging in  (1)  it becomes  p+q+r=0…..(5)  

Also  (3)  simplifies to  pqr+pst+qst+rst=0⟹pqr+(p+q+r)st=0  

⟹pqr=0  

So product of all roots  pqrst=−b=0  

b=0