(Q11)Obtain all other zeros of 3x4+6x3-2x2-10x-5 if two of its zeros are √
5
3
 and - √
5
3

Answer

Let the other two zeros are α and β.

Now prepare the given polynomial 3x4+6x3-2x2-10x-5 with the standard form ax4+bx3+cx2+dx+e we get a=3, b=6, c= -2 d= -10, e= -5

The sum of the zeros =√
5
3
 - √
5
3
= α +β =
-b
a
 =
-
 = -

α+β= -------------(1)

And the product of the zeros=
e
a
(√
5
3
) (-√
5
3
) (α)(β) = -

-
5
3
αβ = -

αβ=

now (α - β)2=(α + β)2 - 4αβ

=(-)2 - (1)

- =

α - β=-----------(2)

Now solving (1) and (2) we get

α + β= -

α - β=

2α= -

α= -, β= -

Then the remaining zeros are - and -