Question

Find all zeroes of polynomial $3x^4+6x^3-2x^2-10x-5$ if two zeroes are $\sqrt{3/5}$ and $-\sqrt{3/5}$

Answer 1

Obtain all zeroes of given polynomial $3x^4+6x^3-2x^2-10x-5$

if two zeroes are $\sqrt{\frac{5}{3}},-\sqrt{\frac{5}{3}}$

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

Since ,

$x=\sqrt{\frac{5}{3}}$ is zero and $x-\sqrt{\frac{5}{3}}=0$ is factor

$x=-\sqrt{\frac{5}{3}}$ is zero and $x+\sqrt{\frac{5}{3}}=0$ is factor

$(x-\sqrt{\frac{5}{3}})(x+\sqrt{\frac{5}{3}})$ is also a factor

$=x^2-\frac{5}{3}$ is also a factor

Now , p(x) divided by $x^2-\frac{5}{3}$ ,we get

$3x^4+6x+3=0$

or $3x^2+3x+3x+3=0$

$3x(x+1)+3(x+1)=0$

$(x+1)(3x+1)=0$

When, $3x+1=0$ Then, $x=-\frac{1}{3}$

When, $x+1=0$ Then., $x=-1$

Hence required zeroes $\frac{-1}{3}and-1$