Find all the zeros of the polynomial $f (x) = 2 x^{4} - 2 x^{3} - 7 x^{2} + 3 x + 6$ , if two of its zeros are $- \sqrt{\frac{3}{2}}$ and $\sqrt{\frac{3}{2}}$

Open in App

Solution

Since, $- \sqrt{\frac{3}{2}}$ and $\sqrt{\frac{3}{2}}$ are zeroes of f(x).
Therefore, $(x + \sqrt{\frac{3}{2}}) (x - \sqrt{\frac{3}{2}}) = (x^{2} - \frac{3}{2}) = \frac{2 x^{2} - 3}{2}$
or $2 x^{2} - 3$ is a factor of f(x).
$∴ 2 x^{4} - 2 x^{3} - 7 x^{2} + 3 x + 6 = (2 x^{2} - 3) (x^{2} - x - 2)$
$= (2 x^{2} - 3) (x - 2) (x + 1)$
$= 2 (x + \sqrt{\frac{3}{2}}) (x - \sqrt{\frac{3}{2}}) (x - 2) (x + 1)$
So, the zeros are $- \sqrt{\frac{3}{2}}, \sqrt{\frac{3}{2}}, 2, - 1$

Suggest Corrections

Similar questions

Q. Find all zeros of the polynomial f(x) = 2x⁴ − 2x³ − 7x² + 3x + 6, if its two zeros are

- \sqrt{\frac{3}{2}} and \sqrt{\frac{3}{2}}

Find all the zeros of the polynomial $(2 x^{4} - 11 x^{3} + 7 x^{2} + 13 x - 7)$ , it being given that two of its zeros are $(3 + \sqrt{2}) a n d (3 - \sqrt{2})$ .

Q. Find all zeros of the polynomial 2x⁴ – 9x³ + 5x² +3x – 1, if two of its zeros are

2 + \sqrt{3} and 2 - \sqrt{3}

Q. Find all the zeroes of the polynomial

2 x^{4} - 11 x^{3} + 7 x^{2} + 13 x - 7

.It being given two of its zeroes are

(3 + \sqrt{2})

and

(3 - \sqrt{2}) .

Q. Find all zeroes of the polynomial

({2 x}^{4} - 9 x^{3} + {5 x}^{2} + 3 x - 1)

if two of its zeroes are

(2 + \sqrt{3}) a n d (2 - \sqrt{3}) .

Join BYJU'S Learning Program

Find all the zeros of the polynomial f(x)=2x4−2x3−7x2+3x+6, if two of its zeros are −√32 and √32

Since, −√32 and √32 are zeroes of f(x).Therefore, (x+√32)(x−√32)=(x2−32)=2x2−32or 2x2−3 is a factor of f(x).∴2x4−2x3−7x2+3x+6=(2x2−3)(x2−x−2)=(2x2−3)(x−2)(x+1)=2(x+√32)(x−√32)(x−2)(x+1)So, the zeros are −√32,√32,2,−1

Find all the zeros of the polynomial $f (x) = 2 x^{4} - 2 x^{3} - 7 x^{2} + 3 x + 6$ , if two of its zeros are $- \sqrt{\frac{3}{2}}$ and $\sqrt{\frac{3}{2}}$