Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
(y+z+y*z/x)*(x+y+x*y/z+x+z+x*z/y)/(2*(x+y+z)+y*z/x+x*y/z+x*z/y)
¿Cómo vas a descomponer esta (y+z+y*z/x)*(x+y+x*y/z+x+z+x*z/y)/(2*(x+y+z)+y*z/x+x*y/z+x*z/y) expresión en fracciones?
Solución
Ha introducido
[src]
/ y*z\ / x*y x*z\
|y + z + ---|*|x + y + --- + x + z + ---|
\ x / \ z y /
-----------------------------------------
y*z x*y x*z
2*(x + y + z) + --- + --- + ---
x z y
$$\frac{\left(\left(y + z\right) + \frac{y z}{x}\right) \left(\left(z + \left(x + \left(\left(x + y\right) + \frac{x y}{z}\right)\right)\right) + \frac{x z}{y}\right)}{\left(\left(2 \left(z + \left(x + y\right)\right) + \frac{y z}{x}\right) + \frac{x y}{z}\right) + \frac{x z}{y}}$$
((y + z + (y*z)/x)*(x + y + (x*y)/z + x + z + (x*z)/y))/(2*(x + y + z) + (y*z)/x + (x*y)/z + (x*z)/y)
Respuesta numérica
[src]
(y + z + y*z/x)*(y + z + 2*x + x*y/z + x*z/y)/(2.0*x + 2.0*y + 2.0*z + x*y/z + x*z/y + y*z/x)
(y + z + y*z/x)*(y + z + 2*x + x*y/z + x*z/y)/(2.0*x + 2.0*y + 2.0*z + x*y/z + x*z/y + y*z/x)
Potencias
[src]
/ y*z\ / x*y x*z\
|y + z + ---|*|y + z + 2*x + --- + ---|
\ x / \ z y /
---------------------------------------
x*y x*z y*z
2*x + 2*y + 2*z + --- + --- + ---
z y x
$$\frac{\left(y + z + \frac{y z}{x}\right) \left(\frac{x y}{z} + 2 x + \frac{x z}{y} + y + z\right)}{\frac{x y}{z} + 2 x + \frac{x z}{y} + 2 y + 2 z + \frac{y z}{x}}$$
(y + z + y*z/x)*(y + z + 2*x + x*y/z + x*z/y)/(2*x + 2*y + 2*z + x*y/z + x*z/y + y*z/x)
Denominador racional
[src]
2 3 2 3 2 3 3 2 3 3 2 2 2 2 2 2
x *y + x *z + y *z + y *z + 2*x*y*z + 2*x*z*y + 3*y*x *z + 3*z*x *y + 4*x*y *z
---------------------------------------------------------------------------------------
2 2 2 2 2 2 2 2 2
x *y + x *z + y *z + 2*x*y*z + 2*x*z*y + 2*y*z*x
$$\frac{x^{2} y^{3} + 3 x^{2} y^{2} z + 3 x^{2} y z^{2} + x^{2} z^{3} + 2 x y^{3} z + 4 x y^{2} z^{2} + 2 x y z^{3} + y^{3} z^{2} + y^{2} z^{3}}{x^{2} y^{2} + 2 x^{2} y z + x^{2} z^{2} + 2 x y^{2} z + 2 x y z^{2} + y^{2} z^{2}}$$
(x^2*y^3 + x^2*z^3 + y^2*z^3 + y^3*z^2 + 2*x*y*z^3 + 2*x*z*y^3 + 3*y*x^2*z^2 + 3*z*x^2*y^2 + 4*x*y^2*z^2)/(x^2*y^2 + x^2*z^2 + y^2*z^2 + 2*x*y*z^2 + 2*x*z*y^2 + 2*y*z*x^2)
Compilar la expresión
[src]
/ / z\\ / / x\ x*z\
|z + y*|1 + -||*|z + 2*x + y*|1 + -| + ---|
\ \ x// \ \ z/ y /
-------------------------------------------
/ x z\ x*z
2*x + 2*z + y*|2 + - + -| + ---
\ z x/ y
$$\frac{\left(y \left(1 + \frac{z}{x}\right) + z\right) \left(2 x + \frac{x z}{y} + y \left(\frac{x}{z} + 1\right) + z\right)}{2 x + \frac{x z}{y} + y \left(\frac{x}{z} + 2 + \frac{z}{x}\right) + 2 z}$$
/ / y\\ / / y\ / x\\
|y + z*|1 + -||*|y + x*|2 + -| + z*|1 + -||
\ \ x// \ \ z/ \ y//
-------------------------------------------
/ y\ / x y\
2*y + x*|2 + -| + z*|2 + - + -|
\ z/ \ y x/
$$\frac{\left(y + z \left(1 + \frac{y}{x}\right)\right) \left(x \left(\frac{y}{z} + 2\right) + y + z \left(\frac{x}{y} + 1\right)\right)}{x \left(\frac{y}{z} + 2\right) + 2 y + z \left(\frac{x}{y} + 2 + \frac{y}{x}\right)}$$
/ / y\\ / / x\ x*y\
|y + z*|1 + -||*|y + 2*x + z*|1 + -| + ---|
\ \ x// \ \ y/ z /
-------------------------------------------
/ x y\ x*y
2*x + 2*y + z*|2 + - + -| + ---
\ y x/ z
$$\frac{\left(y + z \left(1 + \frac{y}{x}\right)\right) \left(\frac{x y}{z} + 2 x + y + z \left(\frac{x}{y} + 1\right)\right)}{\frac{x y}{z} + 2 x + 2 y + z \left(\frac{x}{y} + 2 + \frac{y}{x}\right)}$$
/ / z\\ / / z\ / x\\
|z + y*|1 + -||*|z + x*|2 + -| + y*|1 + -||
\ \ x// \ \ y/ \ z//
-------------------------------------------
/ z\ / x z\
2*z + x*|2 + -| + y*|2 + - + -|
\ y/ \ z x/
$$\frac{\left(y \left(1 + \frac{z}{x}\right) + z\right) \left(x \left(2 + \frac{z}{y}\right) + y \left(\frac{x}{z} + 1\right) + z\right)}{x \left(2 + \frac{z}{y}\right) + y \left(\frac{x}{z} + 2 + \frac{z}{x}\right) + 2 z}$$
/ / z\\ / / x\ / x\\
|z + y*|1 + -||*|2*x + y*|1 + -| + z*|1 + -||
\ \ x// \ \ z/ \ y//
---------------------------------------------
/ x z\ / x\
2*x + y*|2 + - + -| + z*|2 + -|
\ z x/ \ y/
$$\frac{\left(y \left(1 + \frac{z}{x}\right) + z\right) \left(2 x + y \left(\frac{x}{z} + 1\right) + z \left(\frac{x}{y} + 1\right)\right)}{2 x + y \left(\frac{x}{z} + 2 + \frac{z}{x}\right) + z \left(\frac{x}{y} + 2\right)}$$
/ y*z\ / x*y x*z\
|y + z + ---|*|y + z + 2*x + --- + ---|
\ x / \ z y /
---------------------------------------
x*y x*z y*z
2*x + 2*y + 2*z + --- + --- + ---
z y x
$$\frac{\left(y + z + \frac{y z}{x}\right) \left(\frac{x y}{z} + 2 x + \frac{x z}{y} + y + z\right)}{\frac{x y}{z} + 2 x + \frac{x z}{y} + 2 y + 2 z + \frac{y z}{x}}$$
/ / y z\\ / y*z\
|y + z + x*|2 + - + -||*|y + z + ---|
\ \ z y// \ x /
-------------------------------------
/ y z\ y*z
2*y + 2*z + x*|2 + - + -| + ---
\ z y/ x
$$\frac{\left(y + z + \frac{y z}{x}\right) \left(x \left(\frac{y}{z} + 2 + \frac{z}{y}\right) + y + z\right)}{x \left(\frac{y}{z} + 2 + \frac{z}{y}\right) + 2 y + 2 z + \frac{y z}{x}}$$
(y + z + x*(2 + y/z + z/y))*(y + z + y*z/x)/(2*y + 2*z + x*(2 + y/z + z/y) + y*z/x)
Parte trigonométrica
[src]
/ y*z\ / x*y x*z\
|y + z + ---|*|y + z + 2*x + --- + ---|
\ x / \ z y /
---------------------------------------
x*y x*z y*z
2*x + 2*y + 2*z + --- + --- + ---
z y x
$$\frac{\left(y + z + \frac{y z}{x}\right) \left(\frac{x y}{z} + 2 x + \frac{x z}{y} + y + z\right)}{\frac{x y}{z} + 2 x + \frac{x z}{y} + 2 y + 2 z + \frac{y z}{x}}$$
(y + z + y*z/x)*(y + z + 2*x + x*y/z + x*z/y)/(2*x + 2*y + 2*z + x*y/z + x*z/y + y*z/x)
Unión de expresiones racionales
[src]
/ 2 / 2 \\
\x*z + y*\z + x*y + x*z + z*(x + y)//*(x*(y + z) + y*z)
---------------------------------------------------------
/ 2 \ 2 2
y*\y*x + z*(y*z + 2*x*(x + y + z))/ + x *z
$$\frac{\left(x z^{2} + y \left(x y + x z + z^{2} + z \left(x + y\right)\right)\right) \left(x \left(y + z\right) + y z\right)}{x^{2} z^{2} + y \left(x^{2} y + z \left(2 x \left(x + y + z\right) + y z\right)\right)}$$
(x*z^2 + y*(z^2 + x*y + x*z + z*(x + y)))*(x*(y + z) + y*z)/(y*(y*x^2 + z*(y*z + 2*x*(x + y + z))) + x^2*z^2)