Άλγεβρα

Πολυώνυμα

Εύρεση πολυωνύμου με συγκεκριμένη ρίζα

Clear["Global`*"]
MinimalPolynomial[3^(1/2) - 4^(1/3) + 6^(1/5), x]

-15504611567-52841007180 x-21321566145 (x^{2})+45436587400 (x^{3})-9067638735 (x^{4})-15280665768 (x^{5})+17630031015 (x^{6})+7604142480 (x^{7})-13949799615 (x^{8})-377312100 (x^{9})+4361567859 (x^{10})-2182419720 (x^{11})-468640815 (x^{12})+968160240 (x^{13})+74136195 (x^{14})-114140736 (x^{15})-826245 (x^{16})+5523660 (x^{17})-184515 (x^{18})-752040 (x^{19})-441909 (x^{20})+120 (x^{21})+105525 (x^{22})+10800 (x^{23})-11565 (x^{24})-1116 (x^{25})+945 (x^{26})+40 (x^{27})-45 (x^{28})+x^{30}

Διαίρεση πολυωνύμων

Clear["Global`*"]
P[x_] := x^3 - 5 x^2 + 2 x - 1
δ[x_] := x - 3
piliko = PolynomialQuotient[P[x], δ[x], x]
ypoloipo = PolynomialRemainder[P[x], δ[x], x]

-4-2 x+x^{2}

-13

Παραγοντοποίηση-ανάπτυξη πολυωνύμου

Clear["Global`*"]
A = x^3 y^5 - 6 x^2 y^6 + 9 x y^7
B = (x - 2 y)^4 (x + 3 y + 2 x y)^3
Factor[A]
Expand[B]

(x^{3}) (y^{5})-6 (x^{2}) (y^{6})+9 x (y^{7})

({(x-2 y)}^{4}) ({(x+3 y+2 x y)}^{3})

x ({(x-3 y)}^{2}) (y^{5})

x^{7}+(x^{6}) y+6 (x^{7}) y-21 (x^{5}) (y^{2})-12 (x^{6}) (y^{2})+12 (x^{7}) (y^{2})-5 (x^{4}) (y^{3})-90 (x^{5}) (y^{3})-60 (x^{6}) (y^{3})+8 (x^{7}) (y^{3})+160 (x^{3}) (y^{4})+240 (x^{4}) (y^{4})-64 (x^{6}) (y^{4})-72 (x^{2}) (y^{5})+240 (x^{3}) (y^{5})+480 (x^{4}) (y^{5})+192 (x^{5}) (y^{5})-432 x (y^{6})-1152 (x^{2}) (y^{6})-960 (x^{3}) (y^{6})-256 (x^{4}) (y^{6})+432 (y^{7})+864 x (y^{7})+576 (x^{2}) (y^{7})+128 (x^{3}) (y^{7})

Ρητές αλγεβρικές παραστάσεις

Clear["Global`*"]
Together[1/3+1/6]

\frac{1}{2}

Clear["Global`*"]
A = (x^2 - 5 x + 6)/(-4 x + 17 x^2 - 20 x^3 + 4 x^4)
Expand[A]
Apart[A]

\frac{6-5 x+x^{2}}{-4 x+17 (x^{2})-20 (x^{3})+4 (x^{4})}

\frac{6}{-4 x+17 (x^{2})-20 (x^{3})+4 (x^{4})}-\frac{5 x}{-4 x+17 (x^{2})-20 (x^{3})+4 (x^{4})}+\frac{x^{2}}{-4 x+17 (x^{2})-20 (x^{3})+4 (x^{4})}

\frac{1}{98 (-4+x)}-\frac{3}{2 x}-\frac{15}{7 ({(-1+2 x)}^{2})}+\frac{146}{49 (-1+2 x)}

Τριγωνομετρία

Clear["Global`*"]
trig1 = Sin[x + 2 y - 3 w];
trig2 = Sin[x]^3 - Cos[4 x];
(*expands out trigonometric functions inexpr.*)
TrigExpand[trig1]
(*rewrites products and powers of trigonometric functions inexprin \
terms of trigonometric functions with combined arguments.*)
TrigReduce[trig2]
TrigFactor[trig2]

-3 ({\cos(w)}^{2}) \cos(x) ({\cos(y)}^{2}) \sin(w)+\cos(x) ({\cos(y)}^{2}) ({\sin(w)}^{3})+({\cos(w)}^{3}) ({\cos(y)}^{2}) \sin(x)-3 \cos(w) ({\cos(y)}^{2}) ({\sin(w)}^{2}) \sin(x)+2 ({\cos(w)}^{3}) \cos(x) \cos(y) \sin(y)-6 \cos(w) \cos(x) \cos(y) ({\sin(w)}^{2}) \sin(y)+6 ({\cos(w)}^{2}) \cos(y) \sin(w) \sin(x) \sin(y)-2 \cos(y) ({\sin(w)}^{3}) \sin(x) \sin(y)+3 ({\cos(w)}^{2}) \cos(x) \sin(w) ({\sin(y)}^{2})-\cos(x) ({\sin(w)}^{3}) ({\sin(y)}^{2})-({\cos(w)}^{3}) \sin(x) ({\sin(y)}^{2})+3 \cos(w) ({\sin(w)}^{2}) \sin(x) ({\sin(y)}^{2})

(\frac{1}{4}) (-4 \cos(4 x)+3 \sin(x)-\sin(3 x))

-({\sin(\frac{\pi }{4}-\frac{x}{2})}^{2}) (-5+7 \cos(2 x)-10 \sin(x)+4 \sin(3 x))

a = Sin[(x - 5)^3]
Expand[a]
ExpandAll[a]

\sin({(-5+x)}^{3})

\sin({(-5+x)}^{3})

-\sin(125-75 x+15 (x^{2})-x^{3})

Μετατροπή σε εκθετική και αντίστροφα

Clear["Global`*"]
trig1 = Sin[x - 2 y] Cos[x]
TrigToExp[trig1]

\cos(x) \sin(x-2 y)

-\frac{1}{4} I (E^{2 I x-2 I y})+(\frac{1}{4}) I (E^{-2 I x+2 I y})-(\frac{1}{4}) I (E^{-2 I y})+(\frac{1}{4}) I (E^{2 I y})

exp1 = Exp[x^2] - 2^x
ExpToTrig[exp1]

-(2^{x})+E^{x^{2}}

\cosh(x^{2})-\cosh(x \ln(2))+\sinh(x^{2})-\sinh(x \ln(2))

Μιγαδικοί

Clear["Global`*"]
A = Sin[2 + 3 I] Exp[1 - I]
ComplexExpand[A]

z = Sqrt[2]/2 - Sqrt[2]/2 I
Arg[z]
Abs[z]

\frac{1-I}{\sqrt{2}}

-\frac{\pi }{4}

1

z = 2+3I
Conjugate[z]
Re[z]
Im[z]

2+3 I

2-3 I

2

3

ComplexExpand[(k1+k2 I)(L1+L2 I-x-y I)(x+y I)]

k1 L1 x-k2 L2 x-k1 (x^{2})-k2 L1 y-k1 L2 y+2 k2 x y+k1 (y^{2})+I (k2 L1 x+k1 L2 x-k2 (x^{2})+k1 L1 y-k2 L2 y-2 k1 x y+k2 (y^{2}))