To factor the expression, solve the equation where it equals to $0$.
$$2x^{4}+x^{2}-3=0$$
By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-3$ and $q$ divides the leading coefficient $2$. List all candidates $\frac{p}{q}$.
$$±\frac{3}{2},±3,±\frac{1}{2},±1$$
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
$$x=1$$
By Factor theorem, $x-k$ is a factor of the polynomial for each root $k$. Divide $2x^{4}+x^{2}-3$ by $x-1$ to get $2x^{3}+2x^{2}+3x+3$. To factor the result, solve the equation where it equals to $0$.
$$2x^{3}+2x^{2}+3x+3=0$$
By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $3$ and $q$ divides the leading coefficient $2$. List all candidates $\frac{p}{q}$.
$$±\frac{3}{2},±3,±\frac{1}{2},±1$$
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
$$x=-1$$
By Factor theorem, $x-k$ is a factor of the polynomial for each root $k$. Divide $2x^{3}+2x^{2}+3x+3$ by $x+1$ to get $2x^{2}+3$. To factor the result, solve the equation where it equals to $0$.
$$2x^{2}+3=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. Substitute $2$ for $a$, $0$ for $b$, and $3$ for $c$ in the quadratic formula.
