Question

Find all the zeros of 2x4-9x3+5x2+3x-1 if two its zeros are 2+ root 3, 2- root 3​

Answer 1

Answer:

It is given that 2+

3

,2−

3

are two zeroes of the polynomial f(x)=2x

4

−9x

3

+5x

2

+3x−1

∴(x−(2+

3

))(x−(2−

3

))=(x−2−

3

)(x−2+

3

)

=(x−2)

2

−(

3

)

2

=x

2

−4x+4−3

=x

2

−4x+1

∴(x−(2+

3

))(x−(2−

3

))=x

2

−4x+1 is a factor of f(x)

Now divide f(x) by x

2

−4x+1

∴f(x)=(x

2

−4x+1)(2x

2

−x−1)

Hence, the other two zeroes of f(x) are the zeroes of the polynomial 2x

2

−x−1

2x

2

−x−1=(2x+1)(x−1)

Hence, the other two zeroes are −

2

1

,1.