Mister Exam
Factor polynomial x^6-x^4*y^2-x^3+x^2*y
The solution
You have entered
[src]
6 4 2 3 2 x - x *y - x + x *y
$$x^{2} y + \left(- x^{3} + \left(x^{6} - x^{4} y^{2}\right)\right)$$
x^6 - x^4*y^2 - x^3 + x^2*y
Factorization
[src]
/ ________________________________________ \ / ________________________________________ \ / ________________________________________ \
| / ______________________ | | / ______________________ | | / ______________________ |
| / / 2 | | / / 2 | | / / 2 |
| / / / 3\ 6 | | / / / 3\ 6 / ___\ | | / / / 3\ 6 / ___\ |
| / 27 3 \/ \-27 + 2*y / - 4*y | | / 27 3 \/ \-27 + 2*y / - 4*y | 1 I*\/ 3 | | | / 27 3 \/ \-27 + 2*y / - 4*y | 1 I*\/ 3 | |
| 3 / - -- + y + -------------------------- 2 | | 3 / - -- + y + -------------------------- *|- - - -------| 2 | | 3 / - -- + y + -------------------------- *|- - + -------| 2 |
| y \/ 2 2 y | | y \/ 2 2 \ 2 2 / y | | y \/ 2 2 \ 2 2 / y |
x*(x - y)*|x + - + ----------------------------------------------- + -------------------------------------------------|*|x + - + --------------------------------------------------------------- + -----------------------------------------------------------------|*|x + - + --------------------------------------------------------------- + -----------------------------------------------------------------|
| 3 3 ________________________________________| | 3 3 ________________________________________| | 3 3 ________________________________________|
| / ______________________ | | / ______________________ | | / ______________________ |
| / / 2 | | / / 2 | | / / 2 |
| / / / 3\ 6 | | / ___\ / / / 3\ 6 | | / ___\ / / / 3\ 6 |
| / 27 3 \/ \-27 + 2*y / - 4*y | | | 1 I*\/ 3 | / 27 3 \/ \-27 + 2*y / - 4*y | | | 1 I*\/ 3 | / 27 3 \/ \-27 + 2*y / - 4*y |
| 3*3 / - -- + y + -------------------------- | | 3*|- - - -------|*3 / - -- + y + -------------------------- | | 3*|- - + -------|*3 / - -- + y + -------------------------- |
\ \/ 2 2 / \ \ 2 2 / \/ 2 2 / \ \ 2 2 / \/ 2 2 /
$$x \left(x - y\right) \left(x + \left(\frac{y^{2}}{3 \sqrt[3]{y^{3} + \frac{\sqrt{- 4 y^{6} + \left(2 y^{3} - 27\right)^{2}}}{2} - \frac{27}{2}}} + \frac{y}{3} + \frac{\sqrt[3]{y^{3} + \frac{\sqrt{- 4 y^{6} + \left(2 y^{3} - 27\right)^{2}}}{2} - \frac{27}{2}}}{3}\right)\right) \left(x + \left(\frac{y^{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{y^{3} + \frac{\sqrt{- 4 y^{6} + \left(2 y^{3} - 27\right)^{2}}}{2} - \frac{27}{2}}} + \frac{y}{3} + \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{y^{3} + \frac{\sqrt{- 4 y^{6} + \left(2 y^{3} - 27\right)^{2}}}{2} - \frac{27}{2}}}{3}\right)\right) \left(x + \left(\frac{y^{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{y^{3} + \frac{\sqrt{- 4 y^{6} + \left(2 y^{3} - 27\right)^{2}}}{2} - \frac{27}{2}}} + \frac{y}{3} + \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{y^{3} + \frac{\sqrt{- 4 y^{6} + \left(2 y^{3} - 27\right)^{2}}}{2} - \frac{27}{2}}}{3}\right)\right)$$
(((x*(x - y))*(x + y/3 + (-27/2 + y^3 + sqrt((-27 + 2*y^3)^2 - 4*y^6)/2)^(1/3)/3 + y^2/(3*(-27/2 + y^3 + sqrt((-27 + 2*y^3)^2 - 4*y^6)/2)^(1/3))))*(x + y/3 + (-27/2 + y^3 + sqrt((-27 + 2*y^3)^2 - 4*y^6)/2)^(1/3)*(-1/2 - i*sqrt(3)/2)/3 + y^2/(3*(-1/2 - i*sqrt(3)/2)*(-27/2 + y^3 + sqrt((-27 + 2*y^3)^2 - 4*y^6)/2)^(1/3))))*(x + y/3 + (-27/2 + y^3 + sqrt((-27 + 2*y^3)^2 - 4*y^6)/2)^(1/3)*(-1/2 + i*sqrt(3)/2)/3 + y^2/(3*(-1/2 + i*sqrt(3)/2)*(-27/2 + y^3 + sqrt((-27 + 2*y^3)^2 - 4*y^6)/2)^(1/3)))
General simplification
[src]
2 / 4 2 2\ x *\y + x - x - x *y /
$$x^{2} \left(x^{4} - x^{2} y^{2} - x + y\right)$$
x^2*(y + x^4 - x - x^2*y^2)
Assemble expression
[src]
6 3 2 4 2 x - x + y*x - x *y
$$x^{6} - x^{4} y^{2} - x^{3} + x^{2} y$$
x^6 - x^3 + y*x^2 - x^4*y^2
Powers
[src]
6 3 2 4 2 x - x + y*x - x *y
$$x^{6} - x^{4} y^{2} - x^{3} + x^{2} y$$
x^6 - x^3 + y*x^2 - x^4*y^2
Common denominator
[src]
6 3 2 4 2 x - x + y*x - x *y
$$x^{6} - x^{4} y^{2} - x^{3} + x^{2} y$$
x^6 - x^3 + y*x^2 - x^4*y^2
Rational denominator
[src]
6 3 2 4 2 x - x + y*x - x *y
$$x^{6} - x^{4} y^{2} - x^{3} + x^{2} y$$
x^6 - x^3 + y*x^2 - x^4*y^2
Trigonometric part
[src]
6 3 2 4 2 x - x + y*x - x *y
$$x^{6} - x^{4} y^{2} - x^{3} + x^{2} y$$
x^6 - x^3 + y*x^2 - x^4*y^2
Combinatorics
[src]
2 / 3 2\ x *(x - y)*\-1 + x + y*x /
$$x^{2} \left(x - y\right) \left(x^{3} + x^{2} y - 1\right)$$
x^2*(x - y)*(-1 + x^3 + y*x^2)
Combining rational expressions
[src]
2 / / / 2 2\\\ x *\y + x*\-1 + x*\x - y ///
$$x^{2} \left(x \left(x \left(x^{2} - y^{2}\right) - 1\right) + y\right)$$
x^2*(y + x*(-1 + x*(x^2 - y^2)))