Question

Question 3
Obtain all other zeroes of $3x^4+6x^3-2x^2-10x-5$, if two of its zeroes are $\sqrt{\frac{5}{3}}and-\sqrt{\frac{5}{3}}$.

Answer 1

$p\left(x\right)=3x^4+6x^3-2x^2-10x-5$

Since the two zeroes are $\sqrt{\frac{5}{3}}and-\sqrt{\frac{5}{3}}$ ,

$\therefore (x-\sqrt{\frac{5}{3}})(x+\frac{5}{3})=(x^2-\frac{5}{3})isafactorof3x^4+6x^3-2x^2-10x-5$.

Therefore, we divide the given polynomial by $x^2-\frac{5}{3}$.

We factorize $x^2+2x+1$

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

Therefore, its zero is given by x + 1 = 0.

x = -1

As it has the term $(x+1{)}^2$ , therefore, there will be 2 zeroes at x = - 1.

Hence, the zeroes of the given polynomial are $\sqrt{\frac{5}{3}}and-\sqrt{\frac{5}{3}}$ and – 1.