Polynomial Operations and Theorems

Factoring Polynomial Equations

The factored form of a polynomial expression can be used to solve the related equation.
If a polynomial expression can be factored, then the zero product property (if pq=0pq=0, then p=0p=0 or q=0q=0) can be used to solve the related equation that is formed by setting the polynomial expression equal to zero. Each solution will need to be evaluated in the polynomial to determine if it is a true solution.
Step-By-Step Example
Solve a Polynomial Equation by Factoring
Factor to solve the equation:
x4+x36x2=0x^4+x^3-6x^2=0
Step 1

Factor the polynomial.

Start by factoring out the GCF (greatest common factor) of x2x^2.
x4+x36x2=0x2(x2+x6)=0\begin{aligned}x^4+x^3-6x^2&=0\\x^2(x^2+x-6)&=0\end{aligned}
Then factor the resulting trinomial.
x2(x+3)(x2)=0x^2(x+3)(x-2)=0
Step 2

Use the zero product property to identify solutions.

Set each factor equal to zero, and solve.
x2=0x=0x+3=0x=03x=3x2=0x=0+2x=2\boxed{\begin{aligned} x^2&=0\\x&=0\end{aligned}}\;\;\;\;\;\;\;\;\;\;\boxed{\begin{aligned}x+3&=0\\x&=0-3\\x&=-3\end{aligned}}\;\;\;\;\;\;\;\;\;\;\boxed{\begin{aligned}x-2&=0\\x&=0+2\\x&=2\end{aligned}}
Step 3
Check the solutions. Substitute each solution into the original equation.
x4+x36x2=004+036(0)2=?00+00=?00=0x4+x36x2=0(3)4+(3)36(3)2=?0812754=?00=0\boxed{\begin{aligned}x^4+x^3-6x^2&=0\\0^4+0^3-6(0)^2&\stackrel{?}{=}0\\0+0-0&\stackrel{?}{=}0\\0&=0\;\;\checkmark\end{aligned}}\;\;\;\;\;\;\;\;\;\;\boxed{\begin{aligned}x^4+x^3-6x^2&=0\\(-3)^4+(-3)^3-6(-3)^2&\stackrel{?}{=}0\\81-27-54&\stackrel{?}{=}0\\0&=0\;\;\checkmark\end{aligned}}
x4+x36x2=024+236(2)2=?016+824=?00=0\boxed{\begin{aligned}x^4+x^3-6x^2&=0\\2^4+2^3-6(2)^2&\stackrel{?}{=}0\\16+8-24&\stackrel{?}{=}0\\0&=0\;\;\checkmark\end{aligned}}
Solution

The solutions of the equation are x=3x=-3, x=0x=0, and x=2x=2.

The related function of the polynomial is:
f(x)=x4+x36x2f(x)=x^4+x^3-6x^2
The graph of the related function shows that the solutions are the xx-intercepts, which have the ordered pairs (3,0)(-3, 0), (0,0)(0, 0), and (2,0)(2, 0).