Question

If polynomial $f\left(x\right)=x^4-3x^3-x^2+9x-6$ has two zeroes as $-\sqrt{3},\sqrt{3}$ find all zeroes of the polynomial

Answer 1

Since $\sqrt{3}$ and $-\sqrt{3}$ are two zeroes of $f\left(x\right)$.Therefore,

$(x-\sqrt{3})(x+\sqrt{3})=(x^2-3)$ is a factor of $f\left(x\right)$.

On dividing $f\left(x\right)$ by $x^2-3$ we get,

$Quotient=x^2-3x+2$

Therefore $x^2-3x+2$ is also the factor of $f\left(x\right)$.

Then by factorizing it by middle term splitting we get,

$x^2-3x+2$

$=x^2-2x-1x+2$

$=x(x-2)-1(x-2)$

$=(x-2)(x-1)$

Therefore,

$x=2,or,x=1$

Hence other two factors of $f\left(x\right)=1,2$