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Factorizar el polinomio x^4-3*x^2+4

Expresión a simplificar:

Solución

Ha introducido [src]
 4      2    
x  - 3*x  + 4
$$\left(x^{4} - 3 x^{2}\right) + 4$$
x^4 - 3*x^2 + 4
Simplificación general [src]
     4      2
4 + x  - 3*x 
$$x^{4} - 3 x^{2} + 4$$
4 + x^4 - 3*x^2
Factorización [src]
/             /    /  ___\\              /    /  ___\\\ /             /    /  ___\\              /    /  ___\\\ /               /    /  ___\\              /    /  ___\\\ /               /    /  ___\\              /    /  ___\\\
|             |    |\/ 7 ||              |    |\/ 7 ||| |             |    |\/ 7 ||              |    |\/ 7 ||| |               |    |\/ 7 ||              |    |\/ 7 ||| |               |    |\/ 7 ||              |    |\/ 7 |||
|             |atan|-----||              |atan|-----||| |             |atan|-----||              |atan|-----||| |               |atan|-----||              |atan|-----||| |               |atan|-----||              |atan|-----|||
|      ___    |    \  3  /|       ___    |    \  3  /|| |      ___    |    \  3  /|       ___    |    \  3  /|| |        ___    |    \  3  /|       ___    |    \  3  /|| |        ___    |    \  3  /|       ___    |    \  3  /||
|x + \/ 2 *cos|-----------| + I*\/ 2 *sin|-----------||*|x + \/ 2 *cos|-----------| - I*\/ 2 *sin|-----------||*|x + - \/ 2 *cos|-----------| + I*\/ 2 *sin|-----------||*|x + - \/ 2 *cos|-----------| - I*\/ 2 *sin|-----------||
\             \     2     /              \     2     // \             \     2     /              \     2     // \               \     2     /              \     2     // \               \     2     /              \     2     //
$$\left(x + \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)\right)$$
(((x + sqrt(2)*cos(atan(sqrt(7)/3)/2) + i*sqrt(2)*sin(atan(sqrt(7)/3)/2))*(x + sqrt(2)*cos(atan(sqrt(7)/3)/2) - i*sqrt(2)*sin(atan(sqrt(7)/3)/2)))*(x - sqrt(2)*cos(atan(sqrt(7)/3)/2) + i*sqrt(2)*sin(atan(sqrt(7)/3)/2)))*(x - sqrt(2)*cos(atan(sqrt(7)/3)/2) - i*sqrt(2)*sin(atan(sqrt(7)/3)/2))
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(x^{4} - 3 x^{2}\right) + 4$$
Para eso usemos la fórmula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = 1$$
$$b = -3$$
$$c = 4$$
Entonces
$$m = - \frac{3}{2}$$
$$n = \frac{7}{4}$$
Pues,
$$\left(x^{2} - \frac{3}{2}\right)^{2} + \frac{7}{4}$$
Respuesta numérica [src]
4.0 + x^4 - 3.0*x^2
4.0 + x^4 - 3.0*x^2
Unión de expresiones racionales [src]
     2 /      2\
4 + x *\-3 + x /
$$x^{2} \left(x^{2} - 3\right) + 4$$
4 + x^2*(-3 + x^2)
Compilar la expresión [src]
     4      2
4 + x  - 3*x 
$$x^{4} - 3 x^{2} + 4$$
4 + x^4 - 3*x^2
Potencias [src]
     4      2
4 + x  - 3*x 
$$x^{4} - 3 x^{2} + 4$$
4 + x^4 - 3*x^2
Denominador racional [src]
     4      2
4 + x  - 3*x 
$$x^{4} - 3 x^{2} + 4$$
4 + x^4 - 3*x^2
Parte trigonométrica [src]
     4      2
4 + x  - 3*x 
$$x^{4} - 3 x^{2} + 4$$
4 + x^4 - 3*x^2
Combinatoria [src]
     4      2
4 + x  - 3*x 
$$x^{4} - 3 x^{2} + 4$$
4 + x^4 - 3*x^2
Denominador común [src]
     4      2
4 + x  - 3*x 
$$x^{4} - 3 x^{2} + 4$$
4 + x^4 - 3*x^2