Mister Exam
Factor polynomial x^4+4*x^3+8*x^2-5*x-10
The solution
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4 3 2 x + 4*x + 8*x - 5*x - 10
$$\left(- 5 x + \left(8 x^{2} + \left(x^{4} + 4 x^{3}\right)\right)\right) - 10$$
x^4 + 4*x^3 + 8*x^2 - 5*x - 10
General simplification
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4 3 2 -10 + x - 5*x + 4*x + 8*x
$$x^{4} + 4 x^{3} + 8 x^{2} - 5 x - 10$$
-10 + x^4 - 5*x + 4*x^3 + 8*x^2
Factorization
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/ ________________ \ / ________________ \ / ________________ \
| / _______ / ___\ | | / _______ / ___\ | | / _______ |
| / 13 \/ 13785 | 1 I*\/ 3 | 2 | | / 13 \/ 13785 | 1 I*\/ 3 | 2 | | / 13 \/ 13785 2 |
(x + 1)*|x + 1 - 3 / -- + --------- *|- - - -------| + ---------------------------------------|*|x + 1 - 3 / -- + --------- *|- - + -------| + ---------------------------------------|*|x + 1 - 3 / -- + --------- + -----------------------|
| \/ 2 18 \ 2 2 / ________________| | \/ 2 18 \ 2 2 / ________________| | \/ 2 18 ________________|
| / ___\ / _______ | | / ___\ / _______ | | / _______ |
| | 1 I*\/ 3 | / 13 \/ 13785 | | | 1 I*\/ 3 | / 13 \/ 13785 | | / 13 \/ 13785 |
| 3*|- - - -------|*3 / -- + --------- | | 3*|- - + -------|*3 / -- + --------- | | 3*3 / -- + --------- |
\ \ 2 2 / \/ 2 18 / \ \ 2 2 / \/ 2 18 / \ \/ 2 18 /
$$\left(x + 1\right) \left(x + \left(1 + \frac{2}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{13}{2} + \frac{\sqrt{13785}}{18}}} - \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{13}{2} + \frac{\sqrt{13785}}{18}}\right)\right) \left(x + \left(1 - \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{13}{2} + \frac{\sqrt{13785}}{18}} + \frac{2}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{13}{2} + \frac{\sqrt{13785}}{18}}}\right)\right) \left(x + \left(- \sqrt[3]{\frac{13}{2} + \frac{\sqrt{13785}}{18}} + \frac{2}{3 \sqrt[3]{\frac{13}{2} + \frac{\sqrt{13785}}{18}}} + 1\right)\right)$$
(((x + 1)*(x + 1 - (13/2 + sqrt(13785)/18)^(1/3)*(-1/2 - i*sqrt(3)/2) + 2/(3*(-1/2 - i*sqrt(3)/2)*(13/2 + sqrt(13785)/18)^(1/3))))*(x + 1 - (13/2 + sqrt(13785)/18)^(1/3)*(-1/2 + i*sqrt(3)/2) + 2/(3*(-1/2 + i*sqrt(3)/2)*(13/2 + sqrt(13785)/18)^(1/3))))*(x + 1 - (13/2 + sqrt(13785)/18)^(1/3) + 2/(3*(13/2 + sqrt(13785)/18)^(1/3)))
Rational denominator
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4 3 2 -10 + x - 5*x + 4*x + 8*x
$$x^{4} + 4 x^{3} + 8 x^{2} - 5 x - 10$$
-10 + x^4 - 5*x + 4*x^3 + 8*x^2
Powers
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4 3 2 -10 + x - 5*x + 4*x + 8*x
$$x^{4} + 4 x^{3} + 8 x^{2} - 5 x - 10$$
-10 + x^4 - 5*x + 4*x^3 + 8*x^2
Assemble expression
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4 3 2 -10 + x - 5*x + 4*x + 8*x
$$x^{4} + 4 x^{3} + 8 x^{2} - 5 x - 10$$
-10 + x^4 - 5*x + 4*x^3 + 8*x^2
Combining rational expressions
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-10 + x*(-5 + x*(8 + x*(4 + x)))
$$x \left(x \left(x \left(x + 4\right) + 8\right) - 5\right) - 10$$
-10 + x*(-5 + x*(8 + x*(4 + x)))
Common denominator
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4 3 2 -10 + x - 5*x + 4*x + 8*x
$$x^{4} + 4 x^{3} + 8 x^{2} - 5 x - 10$$
-10 + x^4 - 5*x + 4*x^3 + 8*x^2
Numerical answer
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-10.0 + x^4 + 4.0*x^3 + 8.0*x^2 - 5.0*x
-10.0 + x^4 + 4.0*x^3 + 8.0*x^2 - 5.0*x
Trigonometric part
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4 3 2 -10 + x - 5*x + 4*x + 8*x
$$x^{4} + 4 x^{3} + 8 x^{2} - 5 x - 10$$
-10 + x^4 - 5*x + 4*x^3 + 8*x^2
Combinatorics
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/ 3 2 \ (1 + x)*\-10 + x + 3*x + 5*x/
$$\left(x + 1\right) \left(x^{3} + 3 x^{2} + 5 x - 10\right)$$
(1 + x)*(-10 + x^3 + 3*x^2 + 5*x)