Obtain all other zeros of $3 x^{4}+6 x^{3}-2 x^{2}-10 x-5$

Solution:

Two of the zeros of $3 x^{4}+6 x^{3}-2 x^{2}-10 x-5$ are $\sqrt{\frac{5}{3}}$ and $-\sqrt{\frac{5}{3}}$.

$\Rightarrow\left(x-\sqrt{\frac{5}{3}}\right)\left(x+\sqrt{\frac{5}{3}}\right)$

is a factor of the polynomial.

i.e., $\mathrm{x}^{2}-\frac{\mathbf{5}}{\mathbf{3}}$ is a factor.

i.e., $\left(3 x^{2}-5\right)$ is a factor of the polynomial. Then we apply the division algorithm as below:



The other two zeros will be obtained from the quadratic polynomial $q(x)=x^{2}+2 x+1$

Now $x^{2}+2 x+1=(x+1)^{2}$

Its zeros are $-1,-1$.

Hence, all other zeros are $-1,-1$.