Question

if two of the zeros of P(x) = 5 x power 4 - 5 x power 3 -33 x power 2 + 3x-2 root of 3 by 5 and - root 3 by 5 find the other two zeros

Answer 1

a) Zeros of the polynomial :

5x^4-5x^3-33x^2+3x-2=0. (1)

are√3/5 & -√3/5 . (given)

b) Sum of roots of a bi-quadratic polynomial

ax^4+bx^3+cx^2+dx+e=0 is given by

-b/a .
Product of roots = e/a.

c) Let α & β be two other roots of equation :(1)
.

Now,
Sum of roots = -(-5)/5 = 1
√3 / 5 + (-√3/5) + α + β = 1
α+β =1

Product of roots = (-2)/5
α*β*(√3/5)*(-√3/5)= -2/5
α*β = 10/3
Now, here you have product of roots and sum of roots .

d) Make quadratic equation and use quadratic formula to find α and β .

QE is x^2- x+10/3.

x = (1+√1-40/3)/ 2 or (1-√1-40/3)/2.