Math Solve equations Polynomial long division

Polynomial long division

Cubic equations are 3rd degree polynomial equations and have the form:


In order to solve cubic equations and equations of higher degree one needs the Polynomial long division.

First, a zero is guessed by trial and error and then, using polynomial long division, one can transform the equation into a quadratic equation.



  1. Guess a zero $x_N$
  2. Polynomial long division: Divide equation by $(x-x_N)$
  3. Solve the quadratic equation


The quadratic equation obtained after polynomial division can be solved with the quadratic formula.


Solve cubic equation: $x^3-19x-30=0$

  1. Guess a zero

    The first zero must be found by trial and error.
    Use different values for $x$ until 0 results.


    $1^3-19\cdot1-30=-48$ $\neq0$ =>no zero

    $(-1)^3-19\cdot(-1)-30=-12$ $\neq0$ =>no zero

    $(-2)^3-19\cdot(-2)-30=-0$ =>zero at $x_{1}=-2$
  2. Polynomial long division

    The function is divided by $(x-x_1)$. Use polynomial long division for this.


    First calculate $x^3:x$ and write down the result.

    Now $x^2$ is multiplied by $(x+2)$. The result is written in the second line and receives a minus.

    Both lines are now added together with the remainder written underneath.

    Similarly as before, one now calculates $-2x^2:x$. Write the result to the right and multiply it by $(x+2)$ to put it in the line below.

    The two lines are subtracted again.

    Finally, $-15x:x$ is calculated, multiplied back and deducted. The rest is 0 and the polynomial division is done.

  3. Solve quadratic equation

    The new quadratic equation can now be solved e. g. with the pq-formula.

    $x_{2,3} = \frac{p}{2} \pm\sqrt{(\frac{p}{2})^2-q}$

    $x_{2,3} = 1 \pm\sqrt{1^2+15}$
    $x_{2,3} = 1 \pm\sqrt{16}$
    $x_{2,3} = 1 \pm4$

    $x_2=5$ and $x_3=-3$

    The solutions of the initial equation are $x_1=-2$, $x_2=5$ and $x_3=-3$