Calculatrice de factorisation
Factoriser des expressions pas à pas
La calculatrice tentera de factoriser toute expression (polynomiale, binomiale, trinômiale, quadratique, rationnelle, irrationnelle, exponentielle, trigonométrique, ou un mélange de celles-ci), avec les étapes affichées. Pour ce faire, on commence par appliquer certaines substitutions afin de convertir l'expression en polynôme, puis les techniques suivantes sont utilisées : mise en évidence (facteur commun), factorisation des quadratiques, groupement et regroupement, carré d'une somme/d'une différence, cube d'une somme/d'une différence, différence de carrés, somme/différence de cubes, et le théorème des racines rationnelles.
Solution
Your input: factor $$$x^{3} - 8 x^{2} + 20 x - 13$$$.
Since all coefficients are integers, apply the rational zeros theorem.
The trailing coefficient (coefficient of the constant term) is $$$-13$$$.
Find its factors (with plus and minus): $$$\pm 1, \pm 13$$$. These are the possible values for `p`.
The leading coefficient (coefficient of the term with the highest degree) is $$$1$$$.
Find its factors (with plus and minus): $$$\pm 1$$$. These are the possible values for `q`.
Find all possible values of `p/q`: $$$\pm \frac{1}{1}, \pm \frac{13}{1}$$$.
Simplify and remove duplicates (if any): $$$\pm 1, \pm 13$$$.
If `a` is a root of the polynomial `P(x)`, then the remainder from the division of `P(x)` by `x-a` should equal `0`.
Check $$$1$$$: divide $$$x^{3} - 8 x^{2} + 20 x - 13$$$ by $$$x - 1$$$.
The quotient is $$$x^{2} - 7 x + 13$$$, and the remainder is $$$0$$$ (use the synthetic division calculator to see the steps).
Since the remainder is `0`, then $$$1$$$ is the root, and $$$x - 1$$$ is the factor: $$$x^{3} - 8 x^{2} + 20 x - 13 = \left(x - 1\right) \left(x^{2} - 7 x + 13\right)$$$
$${\color{red}{\left(x^{3} - 8 x^{2} + 20 x - 13\right)}} = {\color{red}{\left(x - 1\right) \left(x^{2} - 7 x + 13\right)}}$$
To factor the quadratic function $$$x^{2} - 7 x + 13$$$, we should solve the corresponding quadratic equation $$$x^{2} - 7 x + 13=0$$$.
Indeed, if $$$x_1$$$ and $$$x_2$$$ are the roots of the quadratic equation $$$ax^2+bx+c=0$$$, then $$$ax^2+bx+c=a(x-x_1)(x-x_2)$$$.
Solve the quadratic equation $$$x^{2} - 7 x + 13=0$$$.
The roots are $$$x_{1} = \frac{7}{2} + \frac{\sqrt{3} i}{2}$$$, $$$x_{2} = \frac{7}{2} + \frac{\sqrt{3} i}{2}$$$ (use the quadratic equation calculator to see the steps).
Since the roots are complex, the quadratic can't be factored further (no need to factor using complex numbers). Therefore, we leave $$$x^{2} - 7 x + 13$$$ as it is.
Thus, $$$x^{3} - 8 x^{2} + 20 x - 13=\left(x - 1\right) \left(x^{2} - 7 x + 13\right)$$$.
Answer: $$$x^{3} - 8 x^{2} + 20 x - 13=\left(x - 1\right) \left(x^{2} - 7 x + 13\right)$$$.