To factor the expression, solve the equation where it equals to $0$.
$$x^{6}-41x^{4}+184x^{2}-144=0$$
By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-144$ and $q$ divides the leading coefficient $1$. List all candidates $\frac{p}{q}$.
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 $x^{6}-41x^{4}+184x^{2}-144$ by $x-1$ to get $x^{5}+x^{4}-40x^{3}-40x^{2}+144x+144$. To factor the result, solve the equation where it equals to $0$.
$$x^{5}+x^{4}-40x^{3}-40x^{2}+144x+144=0$$
By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $144$ and $q$ divides the leading coefficient $1$. List all candidates $\frac{p}{q}$.
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 $x^{5}+x^{4}-40x^{3}-40x^{2}+144x+144$ by $x+1$ to get $x^{4}-40x^{2}+144$. To factor the result, solve the equation where it equals to $0$.
$$x^{4}-40x^{2}+144=0$$
By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $144$ and $q$ divides the leading coefficient $1$. List all candidates $\frac{p}{q}$.
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.
$$x=2$$
By Factor theorem, $x-k$ is a factor of the polynomial for each root $k$. Divide $x^{4}-40x^{2}+144$ by $x-2$ to get $x^{3}+2x^{2}-36x-72$. To factor the result, solve the equation where it equals to $0$.
$$x^{3}+2x^{2}-36x-72=0$$
By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-72$ and $q$ divides the leading coefficient $1$. List all candidates $\frac{p}{q}$.
$$±72,±36,±24,±18,±12,±9,±8,±6,±4,±3,±2,±1$$
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.
$$x=-2$$
By Factor theorem, $x-k$ is a factor of the polynomial for each root $k$. Divide $x^{3}+2x^{2}-36x-72$ by $x+2$ to get $x^{2}-36$. To factor the result, solve the equation where it equals to $0$.
$$x^{2}-36=0$$
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 $1$ for $a$, $0$ for $b$, and $-36$ for $c$ in the quadratic formula.
