Mister Exam
Factor polynomial -x^7+3*x^5/16+x^4/16+9*x^3/16+3*x^2/16
The solution
You have entered
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5 4 3 2
7 3*x x 9*x 3*x
- x + ---- + -- + ---- + ----
16 16 16 16
$$\frac{3 x^{2}}{16} + \left(\frac{9 x^{3}}{16} + \left(\frac{x^{4}}{16} + \left(- x^{7} + \frac{3 x^{5}}{16}\right)\right)\right)$$
-x^7 + (3*x^5)/16 + x^4/16 + (9*x^3)/16 + (3*x^2)/16
Fraction decomposition
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-x^7 + x^4/16 + 3*x^2/16 + 3*x^5/16 + 9*x^3/16
$$- x^{7} + \frac{3 x^{5}}{16} + \frac{x^{4}}{16} + \frac{9 x^{3}}{16} + \frac{3 x^{2}}{16}$$
4 2 5 3
7 x 3*x 3*x 9*x
- x + -- + ---- + ---- + ----
16 16 16 16
General simplification
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2 / 2 5 3 \
x *\3 + x - 16*x + 3*x + 9*x/
--------------------------------
16
$$\frac{x^{2} \left(- 16 x^{5} + 3 x^{3} + x^{2} + 9 x + 3\right)}{16}$$
x^2*(3 + x^2 - 16*x^5 + 3*x^3 + 9*x)/16
Factorization
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/ __________________________________________________________________________________________________________________________________________________________\ / __________________________________________________________________________________________________________________________________________________________\ / __________________________________________________________________________________________________________________________________________________________\ / __________________________________________________________________________________________________________________________________________________________ \
| / ____________________ | | / ____________________ | | / ____________________ | | / ____________________ |
| / / _______ | | / / _______ | | / / _______ | | / / _______ |
| / 7 / 9973 \/ 18255 169 15 | | / 7 / 9973 \/ 18255 169 15 | | / 7 / 9973 \/ 18255 169 15 | | / 7 / 9973 \/ 18255 169 15 |
| / - -- - 2*3 / ------ + --------- - ------------------------------ - ---------------------------------------------------------------------------------- | | / - -- - 2*3 / ------ + --------- - ------------------------------ - ---------------------------------------------------------------------------------- | | / - -- - 2*3 / ------ + --------- - ------------------------------ + ---------------------------------------------------------------------------------- | | / - -- - 2*3 / ------ + --------- - ------------------------------ + ---------------------------------------------------------------------------------- |
| _____________________________________________________________________ / 12 \/ 884736 12288 ____________________ _____________________________________________________________________ | | _____________________________________________________________________ / 12 \/ 884736 12288 ____________________ _____________________________________________________________________ | | _____________________________________________________________________ / 12 \/ 884736 12288 ____________________ _____________________________________________________________________ | | / 12 \/ 884736 12288 ____________________ _____________________________________________________________________ _____________________________________________________________________|
| / ____________________ / / _______ / ____________________ | | / ____________________ / / _______ / ____________________ | | / ____________________ / / _______ / ____________________ | | / / _______ / ____________________ / ____________________ |
| / / _______ / / 9973 \/ 18255 / / _______ | | / / _______ / / 9973 \/ 18255 / / _______ | | / / _______ / / 9973 \/ 18255 / / _______ | | / / 9973 \/ 18255 / / _______ / / _______ |
| / 7 / 9973 \/ 18255 169 / 4608*3 / ------ + --------- / 7 / 9973 \/ 18255 169 | | / 7 / 9973 \/ 18255 169 / 4608*3 / ------ + --------- / 7 / 9973 \/ 18255 169 | | / 7 / 9973 \/ 18255 169 / 4608*3 / ------ + --------- / 7 / 9973 \/ 18255 169 | | / 4608*3 / ------ + --------- / 7 / 9973 \/ 18255 169 / 7 / 9973 \/ 18255 169 |
| / - -- + 2*3 / ------ + --------- + ------------------------------ / \/ 884736 12288 16* / - -- + 2*3 / ------ + --------- + ------------------------------ | | / - -- + 2*3 / ------ + --------- + ------------------------------ / \/ 884736 12288 16* / - -- + 2*3 / ------ + --------- + ------------------------------ | | / - -- + 2*3 / ------ + --------- + ------------------------------ / \/ 884736 12288 16* / - -- + 2*3 / ------ + --------- + ------------------------------ | | / \/ 884736 12288 16* / - -- + 2*3 / ------ + --------- + ------------------------------ / - -- + 2*3 / ------ + --------- + ------------------------------ |
| / 24 \/ 884736 12288 ____________________ / / 24 \/ 884736 12288 ____________________ | | / 24 \/ 884736 12288 ____________________ / / 24 \/ 884736 12288 ____________________ | | / 24 \/ 884736 12288 ____________________ / / 24 \/ 884736 12288 ____________________ | | / / 24 \/ 884736 12288 ____________________ / 24 \/ 884736 12288 ____________________ |
| / / _______ / / / _______ | | / / _______ / / / _______ | | / / _______ / / / _______ | | / / / _______ / / _______ |
| / / 9973 \/ 18255 / / / 9973 \/ 18255 | | / / 9973 \/ 18255 / / / 9973 \/ 18255 | | / / 9973 \/ 18255 / / / 9973 \/ 18255 | | / / / 9973 \/ 18255 / / 9973 \/ 18255 |
| / 4608*3 / ------ + --------- / / 4608*3 / ------ + --------- | | / 4608*3 / ------ + --------- / / 4608*3 / ------ + --------- | | / 4608*3 / ------ + --------- / / 4608*3 / ------ + --------- | | / / 4608*3 / ------ + --------- / 4608*3 / ------ + --------- |
| 1 \/ \/ 884736 12288 \/ \/ \/ 884736 12288 | | 1 \/ \/ 884736 12288 \/ \/ \/ 884736 12288 | | 1 \/ \/ 884736 12288 \/ \/ \/ 884736 12288 | | 1 \/ \/ \/ 884736 12288 \/ \/ 884736 12288 |
x*(x - 1)*|x + - - ------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|x + - - ------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|x + - + ------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|x + - - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------|
\ 4 2 2 / \ 4 2 2 / \ 4 2 2 / \ 4 2 2 /
$$x \left(x - 1\right) \left(x + \left(- \frac{\sqrt{- \frac{7}{24} + \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}}}{2} + \frac{1}{4} + \frac{\sqrt{- \frac{15}{16 \sqrt{- \frac{7}{24} + \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}}} - \frac{7}{12} - 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}} - \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}}}}{2}\right)\right) \left(x + \left(- \frac{\sqrt{- \frac{7}{24} + \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}}}{2} + \frac{1}{4} - \frac{\sqrt{- \frac{15}{16 \sqrt{- \frac{7}{24} + \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}}} - \frac{7}{12} - 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}} - \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}}}}{2}\right)\right) \left(x + \left(\frac{\sqrt{- \frac{7}{12} - 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}} - \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}} + \frac{15}{16 \sqrt{- \frac{7}{24} + \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}}}}}{2} + \frac{1}{4} + \frac{\sqrt{- \frac{7}{24} + \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}}}{2}\right)\right) \left(x + \left(- \frac{\sqrt{- \frac{7}{12} - 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}} - \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}} + \frac{15}{16 \sqrt{- \frac{7}{24} + \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}}}}}{2} + \frac{1}{4} + \frac{\sqrt{- \frac{7}{24} + \frac{169}{4608 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{18255}}{12288} + \frac{9973}{884736}}}}{2}\right)\right)$$
((((x*(x - 1))*(x + 1/4 - sqrt(-7/24 + 2*(9973/884736 + sqrt(18255)/12288)^(1/3) + 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)))/2 + sqrt(-7/12 - 2*(9973/884736 + sqrt(18255)/12288)^(1/3) - 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)) - 15/(16*sqrt(-7/24 + 2*(9973/884736 + sqrt(18255)/12288)^(1/3) + 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)))))/2))*(x + 1/4 - sqrt(-7/24 + 2*(9973/884736 + sqrt(18255)/12288)^(1/3) + 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)))/2 - sqrt(-7/12 - 2*(9973/884736 + sqrt(18255)/12288)^(1/3) - 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)) - 15/(16*sqrt(-7/24 + 2*(9973/884736 + sqrt(18255)/12288)^(1/3) + 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)))))/2))*(x + 1/4 + sqrt(-7/24 + 2*(9973/884736 + sqrt(18255)/12288)^(1/3) + 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)))/2 + sqrt(-7/12 - 2*(9973/884736 + sqrt(18255)/12288)^(1/3) - 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)) + 15/(16*sqrt(-7/24 + 2*(9973/884736 + sqrt(18255)/12288)^(1/3) + 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)))))/2))*(x + 1/4 - sqrt(-7/12 - 2*(9973/884736 + sqrt(18255)/12288)^(1/3) - 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)) + 15/(16*sqrt(-7/24 + 2*(9973/884736 + sqrt(18255)/12288)^(1/3) + 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)))))/2 + sqrt(-7/24 + 2*(9973/884736 + sqrt(18255)/12288)^(1/3) + 169/(4608*(9973/884736 + sqrt(18255)/12288)^(1/3)))/2)
Combinatorics
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2 / 2 3 4\
-x *(-1 + x)*\3 + 12*x + 13*x + 16*x + 16*x /
------------------------------------------------
16
$$- \frac{x^{2} \left(x - 1\right) \left(16 x^{4} + 16 x^{3} + 13 x^{2} + 12 x + 3\right)}{16}$$
-x^2*(-1 + x)*(3 + 12*x + 13*x^2 + 16*x^3 + 16*x^4)/16
Rational denominator
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4 7 2 5 3
x - 16*x + 3*x + 3*x + 9*x
-------------------------------
16
$$\frac{- 16 x^{7} + 3 x^{5} + x^{4} + 9 x^{3} + 3 x^{2}}{16}$$
(x^4 - 16*x^7 + 3*x^2 + 3*x^5 + 9*x^3)/16
Combining rational expressions
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2 / / / / 2\\\\
x *\3 + x*\9 + x*\1 + x*\3 - 16*x ////
--------------------------------------
16
$$\frac{x^{2} \left(x \left(x \left(x \left(3 - 16 x^{2}\right) + 1\right) + 9\right) + 3\right)}{16}$$
x^2*(3 + x*(9 + x*(1 + x*(3 - 16*x^2))))/16
Numerical answer
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-x^7 + 0.0625*x^4 + 0.1875*x^2 + 0.1875*x^5 + 0.5625*x^3
-x^7 + 0.0625*x^4 + 0.1875*x^2 + 0.1875*x^5 + 0.5625*x^3
Powers
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4 2 5 3
7 x 3*x 3*x 9*x
- x + -- + ---- + ---- + ----
16 16 16 16
$$- x^{7} + \frac{3 x^{5}}{16} + \frac{x^{4}}{16} + \frac{9 x^{3}}{16} + \frac{3 x^{2}}{16}$$
-x^7 + x^4/16 + 3*x^2/16 + 3*x^5/16 + 9*x^3/16
Common denominator
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4 2 5 3
7 x 3*x 3*x 9*x
- x + -- + ---- + ---- + ----
16 16 16 16
$$- x^{7} + \frac{3 x^{5}}{16} + \frac{x^{4}}{16} + \frac{9 x^{3}}{16} + \frac{3 x^{2}}{16}$$
-x^7 + x^4/16 + 3*x^2/16 + 3*x^5/16 + 9*x^3/16
Assemble expression
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4 2 5 3
7 x 3*x 3*x 9*x
- x + -- + ---- + ---- + ----
16 16 16 16
$$- x^{7} + \frac{3 x^{5}}{16} + \frac{x^{4}}{16} + \frac{9 x^{3}}{16} + \frac{3 x^{2}}{16}$$
-x^7 + x^4/16 + 3*x^2/16 + 3*x^5/16 + 9*x^3/16
Trigonometric part
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4 2 5 3
7 x 3*x 3*x 9*x
- x + -- + ---- + ---- + ----
16 16 16 16
$$- x^{7} + \frac{3 x^{5}}{16} + \frac{x^{4}}{16} + \frac{9 x^{3}}{16} + \frac{3 x^{2}}{16}$$
-x^7 + x^4/16 + 3*x^2/16 + 3*x^5/16 + 9*x^3/16