To factor the expression, solve the equation where it equals to $0$.
$$x^{5}-21x^{3}+16x^{2}+108x-144=0$$
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 $x^{5}-21x^{3}+16x^{2}+108x-144$ by $x-2$ to get $x^{4}+2x^{3}-17x^{2}-18x+72$. To factor the result, solve the equation where it equals to $0$.
$$x^{4}+2x^{3}-17x^{2}-18x+72=0$$
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 $x^{4}+2x^{3}-17x^{2}-18x+72$ by $x-2$ to get $x^{3}+4x^{2}-9x-36$. To factor the result, solve the equation where it equals to $0$.
$$x^{3}+4x^{2}-9x-36=0$$
By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-36$ and $q$ divides the leading coefficient $1$. List all candidates $\frac{p}{q}$.
$$±36,±18,±12,±9,±6,±4,±3,±2,±1$$
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.
$$x=3$$
By Factor theorem, $x-k$ is a factor of the polynomial for each root $k$. Divide $x^{3}+4x^{2}-9x-36$ by $x-3$ to get $x^{2}+7x+12$. To factor the result, solve the equation where it equals to $0$.
$$x^{2}+7x+12=0$$
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 $a$, $7$ for $b$, and $12$ for $c$ in the quadratic formula.
