\documentclass{article} \usepackage{axiom} \setlength{\textwidth}{400pt} \begin{document} \title{\$SPAD/src/input westeralgebra.input} \author{Michael Wester} \maketitle \begin{abstract} These problems come from the web page \begin{verbatim} http://math.unm.edu/~wester/cas_review.html \end{verbatim} \end{abstract} \eject \tableofcontents \eject \begin{chunk}{*} )set break resume )set messages autoload off )set streams calculate 7 )sys rm -f westeralgebra.output )spool westeralgebra.output )clear all \end{chunk} \section{Algebra} One would think that the simplification $2\ 2^n => 2^{(n + 1)}$ would happen automatically or at least easily ... \begin{chunk}{*} --S 1 of 63 2*2**n --R --R --R n --R (1) 2 2 --R Type: Expression(Integer) --E 1 \end{chunk} And how about $4\ 2^n => 2^{(n + 2)}$? [Richard Fateman] \begin{chunk}{*} --S 2 of 63 4*2**n --R --R --R n --R (2) 4 2 --R Type: Expression(Integer) --E 2 \end{chunk} $(-1)^{(n(n + 1))} => 1$ for integer $n$ \begin{chunk}{*} --S 3 of 63 (-1)**(n*(n + 1)) --R --R --R 2 --R n + n --R (3) (- 1) --R Type: Expression(Integer) --E 3 \end{chunk} Also easy $=> 2 (3 x - 5)$ \begin{chunk}{*} --S 4 of 63 factor(6*x - 10) --R --R --R (4) 2(3x - 5) --R Type: Factored(Polynomial(Integer)) --E 4 \end{chunk} Univariate gcd: $gcd(p1, p2) => 1$, $gcd(p1 q, p2 q) => q$ [Richard Liska] \begin{chunk}{*} --S 5 of 63 p1:= 64*x**34 - 21*x**47 - 126*x**8 - 46*x**5 - 16*x**60 - 81 --R --R --R 60 47 34 8 5 --R (5) - 16x - 21x + 64x - 126x - 46x - 81 --R Type: Polynomial(Integer) --E 5 --S 6 of 63 p2:= 72*x**60 - 25*x**25 - 19*x**23 - 22*x**39 - 83*x**52 + 54*x**10 + 81 --R --R --R 60 52 39 25 23 10 --R (6) 72x - 83x - 22x - 25x - 19x + 54x + 81 --R Type: Polynomial(Integer) --E 6 --S 7 of 63 q:= 34*x**19 - 25*x**16 + 70*x**7 + 20*x**3 - 91*x - 86 --R --R --R 19 16 7 3 --R (7) 34x - 25x + 70x + 20x - 91x - 86 --R Type: Polynomial(Integer) --E 7 --S 8 of 63 gcd(p1, p2) --R --R --R (8) 1 --R Type: Polynomial(Integer) --E 8 --S 9 of 63 gcd(expand(p1*q), expand(p2*q)) - q --R --R --R (9) 0 --R Type: Polynomial(Integer) --E 9 \end{chunk} $resultant(p1 q, p2 q) => 0$ \begin{chunk}{*} --S 10 of 63 resultant(expand(p1*q), expand(p2*q), x) --R --R --R (10) 0 --R Type: Polynomial(Integer) --E 10 \end{chunk} How about factorization? $=> p1 * p2$ \begin{chunk}{*} --S 11 of 63 factor(expand(p1 * p2)) --R --R --R (11) --R - --R 60 47 34 8 5 --R (16x + 21x - 64x + 126x + 46x + 81) --R * --R 60 52 39 25 23 10 --R (72x - 83x - 22x - 25x - 19x + 54x + 81) --R Type: Factored(Polynomial(Integer)) --E 11 )clear properties p1 p2 q \end{chunk} Multivariate gcd: $gcd(p1, p2) => 1, gcd(p1 q, p2 q) => q$ \begin{chunk}{*} --S 12 of 63 p1:= 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5 --R --R --R 19 17 5 8 15 9 2 22 --R (12) (24x y - 47x y )z + 6x y z - 3x + 5 --R Type: Polynomial(Integer) --E 12 --S 13 of 63 p2:= 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z --R --R --R 5 8 13 7 7 7 9 16 4 14 --R (13) 34x y z + 20x y z + 12x y z + 80y z --R Type: Polynomial(Integer) --E 13 --S 14 of 63 q:= 11*x**12*y**7*z**13 - 23*x**2*y**8*z**10 + 47*x**17*y**5*z**8 --R --R --R 12 7 13 2 8 10 17 5 8 --R (14) 11x y z - 23x y z + 47x y z --R Type: Polynomial(Integer) --E 14 --S 15 of 63 gcd(p1, p2) --R --R --R (15) 1 --R Type: Polynomial(Integer) --E 15 --S 16 of 63 gcd(expand(p1*q), expand(p2*q)) - q --R --R --R (16) 0 --R Type: Polynomial(Integer) --E 16 \end{chunk} How about factorization? $=> p1 * p2$ \begin{chunk}{*} --S 17 of 63 factor(expand(p1 * p2)) --R --R --R (17) --R 7 19 17 5 8 15 9 2 22 --R 2y z((24x y - 47x y )z + 6x y z - 3x + 5) --R * --R 5 12 7 6 9 9 3 7 --R (17x y z + 10x z + 6x y z + 40y ) --R Type: Factored(Polynomial(Integer)) --E 17 )clear properties p1 p2 q \end{chunk} $=> x^n {\textrm\ for\ } n > 0$ [Chris Hurlburt] \begin{chunk}{*} --S 18 of 63 gcd(2*x**(n + 4) - x**(n + 2), 4*x**(n + 1) + 3*x**n) --R --R --R (18) 1 --R Type: Expression(Integer) --E 18 \end{chunk} Resultants. If the resultant of two polynomials is zero, this implies they have a common factor. See Keith O. Geddes, Stephen R. Czapor and George Labahn, ``Algorithms for Computer Algebra'', Kluwer Academic Publishers, 1992, p. 286 $=> 0$ \begin{chunk}{*} --S 19 of 63 resultant(3*x**4 + 3*x**3 + x**2 - x - 2, x**3 - 3*x**2 + x + 5, x) --R --R --R (19) 0 --R Type: Polynomial(Integer) --E 19 \end{chunk} Numbers are nice, but symbols allow for variability---try some high school algebra: rational simplification $=> (x - 2)/(x + 2)$ \begin{chunk}{*} --S 20 of 63 (x**2 - 4)/(x**2 + 4*x + 4) --R --R --R x - 2 --R (20) ----- --R x + 2 --R Type: Fraction(Polynomial(Integer)) --E 20 \end{chunk} This example requires more sophistication $=> e^{(x/2)} - 1$ \begin{chunk}{*} --S 21 of 63 [(%e**x - 1)/(%e**(x/2) + 1), (exp(x) - 1)/(exp(x/2) + 1)] --R --R --R x x --R %e - 1 %e - 1 --R (21) [-------,-------] --R x x --R - - --R 2 2 --R %e + 1 %e + 1 --R Type: List(Expression(Integer)) --E 21 --S 22 of 63 map(normalize, %) --R --R --R x x --R - - --R 2 2 --R (22) [%e - 1,%e - 1] --R Type: List(Expression(Integer)) --E 22 \end{chunk} Expand and factor polynomials \begin{chunk}{*} --S 23 of 63 (x + 1)**20 --R --R --R (23) --R 20 19 18 17 16 15 14 13 --R x + 20x + 190x + 1140x + 4845x + 15504x + 38760x + 77520x --R + --R 12 11 10 9 8 7 6 --R 125970x + 167960x + 184756x + 167960x + 125970x + 77520x + 38760x --R + --R 5 4 3 2 --R 15504x + 4845x + 1140x + 190x + 20x + 1 --R Type: Polynomial(Integer) --E 23 --S 24 of 63 D(%, x) --R --R --R (24) --R 19 18 17 16 15 14 13 --R 20x + 380x + 3420x + 19380x + 77520x + 232560x + 542640x --R + --R 12 11 10 9 8 7 --R 1007760x + 1511640x + 1847560x + 1847560x + 1511640x + 1007760x --R + --R 6 5 4 3 2 --R 542640x + 232560x + 77520x + 19380x + 3420x + 380x + 20 --R Type: Polynomial(Integer) --E 24 --S 25 of 63 factor(%) --R --R --R 19 --R (25) 20(x + 1) --R Type: Factored(Polynomial(Integer)) --E 25 \end{chunk} Completely factor this polynomial, then try to multiply it back together! \begin{chunk}{*} --S 26 of 63 radicalSolve(x**3 + x**2 - 7 = 0, x) --R --R --R (26) --R [ --R x = --R +------------------+2 --R | +----+ +-+ --R +---+ |9\|1295 + 187\|3 --R (- 9\|- 3 + 9) |------------------ --R 3| +-+ --R \| 54\|3 --R + --R +------------------+ --R | +----+ +-+ --R +---+ |9\|1295 + 187\|3 --R (- 3\|- 3 - 3) |------------------ - 2 --R 3| +-+ --R \| 54\|3 --R / --R +------------------+ --R | +----+ +-+ --R +---+ |9\|1295 + 187\|3 --R (9\|- 3 + 9) |------------------ --R 3| +-+ --R \| 54\|3 --R , --R --R x = --R +------------------+2 --R | +----+ +-+ --R +---+ |9\|1295 + 187\|3 --R (- 9\|- 3 - 9) |------------------ --R 3| +-+ --R \| 54\|3 --R + --R +------------------+ --R | +----+ +-+ --R +---+ |9\|1295 + 187\|3 --R (- 3\|- 3 + 3) |------------------ + 2 --R 3| +-+ --R \| 54\|3 --R / --R +------------------+ --R | +----+ +-+ --R +---+ |9\|1295 + 187\|3 --R (9\|- 3 - 9) |------------------ --R 3| +-+ --R \| 54\|3 --R , --R +------------------+2 +------------------+ --R | +----+ +-+ | +----+ +-+ --R |9\|1295 + 187\|3 |9\|1295 + 187\|3 --R 9 |------------------ - 3 |------------------ + 1 --R 3| +-+ 3| +-+ --R \| 54\|3 \| 54\|3 --R x= ----------------------------------------------------] --R +------------------+ --R | +----+ +-+ --R |9\|1295 + 187\|3 --R 9 |------------------ --R 3| +-+ --R \| 54\|3 --R Type: List(Equation(Expression(Integer))) --E 26 --S 27 of 63 reduce(*, map(e +-> lhs(e) - rhs(e), %)) --R --R --R 3 2 +-+ +----+ 3 2 --R (9x + 9x - 63)\|3 \|1295 + 561x + 561x - 3927 --R (27) -------------------------------------------------- --R +-+ +----+ +-+2 --R 9\|3 \|1295 + 187\|3 --R Type: Expression(Integer) --E 27 --S 28 of 63 x**100 - 1 --R --R --R 100 --R (28) x - 1 --R Type: Polynomial(Integer) --E 28 --S 29 of 63 factor(%) --R --R --R (29) --R 2 4 3 2 4 3 2 --R (x - 1)(x + 1)(x + 1)(x - x + x - x + 1)(x + x + x + x + 1) --R * --R 8 6 4 2 20 15 10 5 20 15 10 5 --R (x - x + x - x + 1)(x - x + x - x + 1)(x + x + x + x + 1) --R * --R 40 30 20 10 --R (x - x + x - x + 1) --R Type: Factored(Polynomial(Integer)) --E 29 \end{chunk} Factorization over the complex rationals $=> (2 x + 3 i) (2 x - 3 i) (x + 1 + 4 i) (x + 1 - 4 i)$ \begin{chunk}{*} --S 30 of 63 factor(4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153, [rootOf(i**2 + 1)]) --R --R --R 3i 3i --R (30) 4(x - 4i + 1)(x - --)(x + --)(x + 4i + 1) --R 2 2 --R Type: Factored(Polynomial(AlgebraicNumber)) --E 30 \end{chunk} Algebraic extensions \begin{chunk}{*} --S 31 of 63 sqrt2:= rootOf(sqrt2**2 - 2) --R --R --R (31) sqrt2 --R Type: AlgebraicNumber --E 31 \end{chunk} $=> sqrt2 + 1$ \begin{chunk}{*} --S 32 of 63 1/(sqrt2 - 1) --R --R --R (32) sqrt2 + 1 --R Type: AlgebraicNumber --E 32 \end{chunk} $=> (x^2 - 2 x - 3)/(x - sqrt2) = (x + 1) (x - 3)/(x - sqrt2)$ [Richard Liska] \begin{chunk}{*} --S 33 of 63 (x**3 + (sqrt2 - 2)*x**2 - (2*sqrt2 + 3)*x - 3*sqrt2)/(x**2 - 2) --R --R --R 2 --R x - 2x - 3 --R (33) ----------- --R x - sqrt2 --R Type: Fraction(Polynomial(AlgebraicNumber)) --E 33 --S 34 of 63 numer(%)/ratDenom(denom(%)) --R --R --R 2 --R - x + 2x + 3 --R (34) ------------- --R sqrt2 - x --R Type: Expression(Integer) --E 34 )clear properties sqrt2 \end{chunk} Multiple algebraic extensions \begin{chunk}{*} --S 35 of 63 sqrt3:= rootOf(sqrt3**2 - 3) --R --R --R (35) sqrt3 --R Type: AlgebraicNumber --E 35 --S 36 of 63 cbrt2:= rootOf(cbrt2**3 - 2) --R --R --R (36) cbrt2 --R Type: AlgebraicNumber --E 36 \end{chunk} $=> 2 cbrt2 + 8 sqrt3 + 18 cbrt2^2 + 12 cbrt2 sqrt3 + 9$ \begin{chunk}{*} --S 37 of 63 (cbrt2 + sqrt3)**4 --R --R --R 2 --R (37) (12cbrt2 + 8)sqrt3 + 18cbrt2 + 2cbrt2 + 9 --R Type: AlgebraicNumber --E 37 )clear properties sqrt3 cbrt2 \end{chunk} Factor polynomials over finite fields and field extensions \begin{chunk}{*} --S 38 of 63 p:= x**4 - 3*x**2 + 1 --R --R --R 4 2 --R (38) x - 3x + 1 --R Type: Polynomial(Integer) --E 38 --S 39 of 63 factor(p) --R --R --R 2 2 --R (39) (x - x - 1)(x + x - 1) --R Type: Factored(Polynomial(Integer)) --E 39 \end{chunk} $=> (x - 2)^2 (x + 2)^2 {\textrm\ mod\ } 5$ \begin{chunk}{*} --S 40 of 63 factor(p :: Polynomial(PrimeField(5))) --R --R --R 2 2 --R (40) (x + 2) (x + 3) --R Type: Factored(Polynomial(PrimeField(5))) --E 40 --S 41 of 63 expand(%) --R --R --R 4 2 --R (41) x + 2x + 1 --R Type: Polynomial(PrimeField(5)) --E 41 \end{chunk} $=> (x^2 + x + 1) (x^9 - x^8 + x^6 - x^5 + x^3 - x^2 + 1){\textrm\ mod\ } 65537$ [Paul Zimmermann] \begin{chunk}{*} --S 42 of 63 factor(x**11 + x + 1 :: Polynomial(PrimeField(65537))) --R --R --R 2 9 8 6 5 3 2 --R (42) (x + x + 1)(x + 65536x + x + 65536x + x + 65536x + 1) --R Type: Factored(Polynomial(PrimeField(65537))) --E 42 \end{chunk} $=> (x - phi) (x + phi) (x - phi + 1) (x + phi - 1)$ where $phi^2 - phi - 1 = 0$ or $phi = (1 \pm sqrt(5))/2$ \begin{chunk}{*} --S 43 of 63 phi:= rootOf(phi**2 - phi - 1) --R --R --R (43) phi --R Type: AlgebraicNumber --E 43 --S 44 of 63 factor(p, [phi]) --R --R --R (44) (x - phi)(x - phi + 1)(x + phi - 1)(x + phi) --R Type: Factored(Polynomial(AlgebraicNumber)) --E 44 )clear properties phi p --S 45 of 63 expand((x - 2*y**2 + 3*z**3)**20) --R --R --R (45) --R 60 2 57 --R 3486784401z + (- 46490458680y + 23245229340x)z --R + --R 4 2 2 54 --R (294439571640y - 294439571640x y + 73609892910x )z --R + --R 6 4 2 2 --R - 1177758286560y + 1766637429840x y - 883318714920x y --R + --R 3 --R 147219785820x --R * --R 51 --R z --R + --R 8 6 2 4 --R 3336981811920y - 6673963623840x y + 5005472717880x y --R + --R 3 2 4 --R - 1668490905960x y + 208561363245x --R * --R 48 --R z --R + --R 10 8 2 6 --R - 7118894532096y + 17797236330240x y - 17797236330240x y --R + --R 3 4 4 2 5 --R 8898618165120x y - 2224654541280x y + 222465454128x --R * --R 45 --R z --R + --R 12 10 2 8 --R 11864824220160y - 35594472660480x y + 44493090825600x y --R + --R 3 6 4 4 5 2 --R - 29662060550400x y + 11123272706400x y - 2224654541280x y --R + --R 6 --R 185387878440x --R * --R 42 --R z --R + --R 14 12 2 10 --R - 15819765626880y + 55369179694080x y - 83053769541120x y --R + --R 3 8 4 6 5 4 --R 69211474617600x y - 34605737308800x y + 10381721192640x y --R + --R 6 2 7 --R - 1730286865440x y + 123591918960x --R * --R 39 --R z --R + --R 16 14 2 12 --R 17138079429120y - 68552317716480x y + 119966556003840x y --R + --R 3 10 4 8 5 6 --R - 119966556003840x y + 74979097502400x y - 29991639000960x y --R + --R 6 4 7 2 8 --R 7497909750240x y - 1071129964320x y + 66945622770x --R * --R 36 --R z --R + --R 18 16 2 14 --R - 15233848381440y + 68552317716480x y - 137104635432960x y --R + --R 3 12 4 10 5 8 --R 159955408005120x y - 119966556003840x y + 59983278001920x y --R + --R 6 6 7 4 8 2 --R - 19994426000640x y + 4284519857280x y - 535564982160x y --R + --R 9 --R 29753610120x --R * --R 33 --R z --R + --R 20 18 2 16 --R 11171488813056y - 55857444065280x y + 125679249146880x y --R + --R 3 14 4 12 5 10 --R - 167572332195840x y + 146625790671360x y - 87975474402816x y --R + --R 6 8 7 6 8 4 --R 36656447667840x y - 10473270762240x y + 1963738267920x y --R + --R 9 2 10 --R - 218193140880x y + 10909657044x --R * --R 30 --R z --R + --R 22 20 2 18 --R - 6770599280640y + 37238296043520x y - 93095740108800x y --R + --R 3 16 4 14 5 12 --R 139643610163200x y - 139643610163200x y + 97750527114240x y --R + --R 6 10 7 8 8 6 --R - 48875263557120x y + 17455451270400x y - 4363862817600x y --R + --R 9 4 10 2 11 --R 727310469600x y - 72731046960x y + 3305956680x --R * --R 27 --R z --R + --R 24 22 2 20 --R 3385299640320y - 20311797841920x y + 55857444065280x y --R + --R 3 18 4 16 5 14 --R - 93095740108800x y + 104732707622400x y - 83786166097920x y --R + --R 6 12 7 10 8 8 --R 48875263557120x y - 20946541524480x y + 6545794226400x y --R + --R 9 6 10 4 11 2 12 --R - 1454620939200x y + 218193140880x y - 19835740080x y + 826489170x --R * --R 24 --R z --R + --R 26 24 2 22 --R - 1388840878080y + 9027465707520x y - 27082397122560x y --R + --R 3 20 4 18 5 16 --R 49651061391360x y - 62063826739200x y + 55857444065280x y --R + --R 6 14 7 12 8 10 --R - 37238296043520x y + 18619148021760x y - 6982180508160x y --R + --R 9 8 10 6 11 4 --R 1939494585600x y - 387898917120x y + 52895306880x y --R + --R 12 2 13 --R - 4407942240x y + 169536240x --R * --R 21 --R z --R + --R 28 26 2 24 --R 462946959360y - 3240628715520x y + 10532043325440x y --R + --R 3 22 4 20 5 18 --R - 21064086650880x y + 28963119144960x y - 28963119144960x y --R + --R 6 16 7 14 8 12 --R 21722339358720x y - 12412765347840x y + 5430584839680x y --R + --R 9 10 10 8 11 6 --R - 1810194946560x y + 452548736640x y - 82281588480x y --R + --R 12 4 13 2 14 --R 10285198560x y - 791169120x y + 28256040x --R * --R 18 --R z --R + --R 30 28 2 26 --R - 123452522496y + 925893918720x y - 3240628715520x y --R + --R 3 24 4 22 5 20 --R 7021362216960x y - 10532043325440x y + 11585247657984x y --R + --R 6 18 7 16 8 14 --R - 9654373048320x y + 6206382673920x y - 3103191336960x y --R + --R 9 12 10 10 11 8 --R 1206796631040x y - 362038989312x y + 82281588480x y --R + --R 12 6 13 4 14 2 15 --R - 13713598080x y + 1582338240x y - 113024160x y + 3767472x --R * --R 15 --R z --R + --R 32 30 2 28 --R 25719275520y - 205754204160x y + 771578265600x y --R + --R 3 26 4 24 5 22 --R - 1800349286400x y + 2925567590400x y - 3510681108480x y --R + --R 6 20 7 18 8 16 --R 3218124349440x y - 2298660249600x y + 1292996390400x y --R + --R 9 14 10 12 11 10 --R - 574665062400x y + 201132771840x y - 54854392320x y --R + --R 12 8 13 6 14 4 15 2 --R 11427998400x y - 1758153600x y + 188373600x y - 12558240x y --R + --R 16 --R 392445x --R * --R 12 --R z --R + --R 34 32 2 30 --R - 4034396160y + 34292367360x y - 137169469440x y --R + --R 3 28 4 26 5 24 --R 342923673600x y - 600116428800x y + 780151357440x y --R + --R 6 22 7 20 8 18 --R - 780151357440x y + 612976066560x y - 383110041600x y --R + --R 9 16 10 14 11 12 --R 191555020800x y - 76622008320x y + 24379729920x y --R + --R 12 10 13 8 14 6 15 4 --R - 6094932480x y + 1172102400x y - 167443200x y + 16744320x y --R + --R 16 2 17 --R - 1046520x y + 30780x --R * --R 9 --R z --R + --R 36 34 2 32 3 30 --R 448266240y - 4034396160x y + 17146183680x y - 45723156480x y --R + --R 4 28 5 26 6 24 --R 85730918400x y - 120023285760x y + 130025226240x y --R + --R 7 22 8 20 9 18 --R - 111450193920x y + 76622008320x y - 42567782400x y --R + --R 10 16 11 14 12 12 --R 19155502080x y - 6965637120x y + 2031644160x y --R + --R 13 10 14 8 15 6 16 4 --R - 468840960x y + 83721600x y - 11162880x y + 1046520x y --R + --R 17 2 18 --R - 61560x y + 1710x --R * --R 6 --R z --R + --R 38 36 2 34 3 32 --R - 31457280y + 298844160x y - 1344798720x y + 3810263040x y --R + --R 4 30 5 28 6 26 --R - 7620526080x y + 11430789120x y - 13335920640x y --R + --R 7 24 8 22 9 20 10 18 --R 12383354880x y - 9287516160x y + 5675704320x y - 2837852160x y --R + --R 11 16 12 14 13 12 14 10 --R 1160939520x y - 386979840x y + 104186880x y - 22325760x y --R + --R 15 8 16 6 17 4 18 2 19 --R 3720960x y - 465120x y + 41040x y - 2280x y + 60x --R * --R 3 --R z --R + --R 40 38 2 36 3 34 --R 1048576y - 10485760x y + 49807360x y - 149422080x y --R + --R 4 32 5 30 6 28 7 26 --R 317521920x y - 508035072x y + 635043840x y - 635043840x y --R + --R 8 24 9 22 10 20 11 18 --R 515973120x y - 343982080x y + 189190144x y - 85995520x y --R + --R 12 16 13 14 14 12 15 10 16 8 --R 32248320x y - 9922560x y + 2480640x y - 496128x y + 77520x y --R + --R 17 6 18 4 19 2 20 --R - 9120x y + 760x y - 40x y + x --R Type: Polynomial(Integer) --E 45 --S 46 of 63 factor(%) --R --R --R 3 2 20 --R (46) (3z - 2y + x) --R Type: Factored(Polynomial(Integer)) --E 46 --S 47 of 63 expand((sin(x) - 2*cos(y)**2 + 3*tan(z)**3)**20) --R --R --R (47) --R 60 2 57 --R 3486784401tan(z) + (23245229340sin(x) - 46490458680cos(y) )tan(z) --R + --R 2 2 4 --R (73609892910sin(x) - 294439571640cos(y) sin(x) + 294439571640cos(y) ) --R * --R 54 --R tan(z) --R + --R 3 2 2 --R 147219785820sin(x) - 883318714920cos(y) sin(x) --R + --R 4 6 --R 1766637429840cos(y) sin(x) - 1177758286560cos(y) --R * --R 51 --R tan(z) --R + --R 4 2 3 --R 208561363245sin(x) - 1668490905960cos(y) sin(x) --R + --R 4 2 6 --R 5005472717880cos(y) sin(x) - 6673963623840cos(y) sin(x) --R + --R 8 --R 3336981811920cos(y) --R * --R 48 --R tan(z) --R + --R 5 2 4 --R 222465454128sin(x) - 2224654541280cos(y) sin(x) --R + --R 4 3 6 2 --R 8898618165120cos(y) sin(x) - 17797236330240cos(y) sin(x) --R + --R 8 10 --R 17797236330240cos(y) sin(x) - 7118894532096cos(y) --R * --R 45 --R tan(z) --R + --R 6 2 5 --R 185387878440sin(x) - 2224654541280cos(y) sin(x) --R + --R 4 4 6 3 --R 11123272706400cos(y) sin(x) - 29662060550400cos(y) sin(x) --R + --R 8 2 10 --R 44493090825600cos(y) sin(x) - 35594472660480cos(y) sin(x) --R + --R 12 --R 11864824220160cos(y) --R * --R 42 --R tan(z) --R + --R 7 2 6 --R 123591918960sin(x) - 1730286865440cos(y) sin(x) --R + --R 4 5 6 4 --R 10381721192640cos(y) sin(x) - 34605737308800cos(y) sin(x) --R + --R 8 3 10 2 --R 69211474617600cos(y) sin(x) - 83053769541120cos(y) sin(x) --R + --R 12 14 --R 55369179694080cos(y) sin(x) - 15819765626880cos(y) --R * --R 39 --R tan(z) --R + --R 8 2 7 --R 66945622770sin(x) - 1071129964320cos(y) sin(x) --R + --R 4 6 6 5 --R 7497909750240cos(y) sin(x) - 29991639000960cos(y) sin(x) --R + --R 8 4 10 3 --R 74979097502400cos(y) sin(x) - 119966556003840cos(y) sin(x) --R + --R 12 2 14 --R 119966556003840cos(y) sin(x) - 68552317716480cos(y) sin(x) --R + --R 16 --R 17138079429120cos(y) --R * --R 36 --R tan(z) --R + --R 9 2 8 --R 29753610120sin(x) - 535564982160cos(y) sin(x) --R + --R 4 7 6 6 --R 4284519857280cos(y) sin(x) - 19994426000640cos(y) sin(x) --R + --R 8 5 10 4 --R 59983278001920cos(y) sin(x) - 119966556003840cos(y) sin(x) --R + --R 12 3 14 2 --R 159955408005120cos(y) sin(x) - 137104635432960cos(y) sin(x) --R + --R 16 18 --R 68552317716480cos(y) sin(x) - 15233848381440cos(y) --R * --R 33 --R tan(z) --R + --R 10 2 9 --R 10909657044sin(x) - 218193140880cos(y) sin(x) --R + --R 4 8 6 7 --R 1963738267920cos(y) sin(x) - 10473270762240cos(y) sin(x) --R + --R 8 6 10 5 --R 36656447667840cos(y) sin(x) - 87975474402816cos(y) sin(x) --R + --R 12 4 14 3 --R 146625790671360cos(y) sin(x) - 167572332195840cos(y) sin(x) --R + --R 16 2 18 --R 125679249146880cos(y) sin(x) - 55857444065280cos(y) sin(x) --R + --R 20 --R 11171488813056cos(y) --R * --R 30 --R tan(z) --R + --R 11 2 10 --R 3305956680sin(x) - 72731046960cos(y) sin(x) --R + --R 4 9 6 8 --R 727310469600cos(y) sin(x) - 4363862817600cos(y) sin(x) --R + --R 8 7 10 6 --R 17455451270400cos(y) sin(x) - 48875263557120cos(y) sin(x) --R + --R 12 5 14 4 --R 97750527114240cos(y) sin(x) - 139643610163200cos(y) sin(x) --R + --R 16 3 18 2 --R 139643610163200cos(y) sin(x) - 93095740108800cos(y) sin(x) --R + --R 20 22 --R 37238296043520cos(y) sin(x) - 6770599280640cos(y) --R * --R 27 --R tan(z) --R + --R 12 2 11 --R 826489170sin(x) - 19835740080cos(y) sin(x) --R + --R 4 10 6 9 --R 218193140880cos(y) sin(x) - 1454620939200cos(y) sin(x) --R + --R 8 8 10 7 --R 6545794226400cos(y) sin(x) - 20946541524480cos(y) sin(x) --R + --R 12 6 14 5 --R 48875263557120cos(y) sin(x) - 83786166097920cos(y) sin(x) --R + --R 16 4 18 3 --R 104732707622400cos(y) sin(x) - 93095740108800cos(y) sin(x) --R + --R 20 2 22 --R 55857444065280cos(y) sin(x) - 20311797841920cos(y) sin(x) --R + --R 24 --R 3385299640320cos(y) --R * --R 24 --R tan(z) --R + --R 13 2 12 --R 169536240sin(x) - 4407942240cos(y) sin(x) --R + --R 4 11 6 10 --R 52895306880cos(y) sin(x) - 387898917120cos(y) sin(x) --R + --R 8 9 10 8 --R 1939494585600cos(y) sin(x) - 6982180508160cos(y) sin(x) --R + --R 12 7 14 6 --R 18619148021760cos(y) sin(x) - 37238296043520cos(y) sin(x) --R + --R 16 5 18 4 --R 55857444065280cos(y) sin(x) - 62063826739200cos(y) sin(x) --R + --R 20 3 22 2 --R 49651061391360cos(y) sin(x) - 27082397122560cos(y) sin(x) --R + --R 24 26 --R 9027465707520cos(y) sin(x) - 1388840878080cos(y) --R * --R 21 --R tan(z) --R + --R 14 2 13 --R 28256040sin(x) - 791169120cos(y) sin(x) --R + --R 4 12 6 11 --R 10285198560cos(y) sin(x) - 82281588480cos(y) sin(x) --R + --R 8 10 10 9 --R 452548736640cos(y) sin(x) - 1810194946560cos(y) sin(x) --R + --R 12 8 14 7 --R 5430584839680cos(y) sin(x) - 12412765347840cos(y) sin(x) --R + --R 16 6 18 5 --R 21722339358720cos(y) sin(x) - 28963119144960cos(y) sin(x) --R + --R 20 4 22 3 --R 28963119144960cos(y) sin(x) - 21064086650880cos(y) sin(x) --R + --R 24 2 26 --R 10532043325440cos(y) sin(x) - 3240628715520cos(y) sin(x) --R + --R 28 --R 462946959360cos(y) --R * --R 18 --R tan(z) --R + --R 15 2 14 4 13 --R 3767472sin(x) - 113024160cos(y) sin(x) + 1582338240cos(y) sin(x) --R + --R 6 12 8 11 --R - 13713598080cos(y) sin(x) + 82281588480cos(y) sin(x) --R + --R 10 10 12 9 --R - 362038989312cos(y) sin(x) + 1206796631040cos(y) sin(x) --R + --R 14 8 16 7 --R - 3103191336960cos(y) sin(x) + 6206382673920cos(y) sin(x) --R + --R 18 6 20 5 --R - 9654373048320cos(y) sin(x) + 11585247657984cos(y) sin(x) --R + --R 22 4 24 3 --R - 10532043325440cos(y) sin(x) + 7021362216960cos(y) sin(x) --R + --R 26 2 28 --R - 3240628715520cos(y) sin(x) + 925893918720cos(y) sin(x) --R + --R 30 --R - 123452522496cos(y) --R * --R 15 --R tan(z) --R + --R 16 2 15 4 14 --R 392445sin(x) - 12558240cos(y) sin(x) + 188373600cos(y) sin(x) --R + --R 6 13 8 12 --R - 1758153600cos(y) sin(x) + 11427998400cos(y) sin(x) --R + --R 10 11 12 10 --R - 54854392320cos(y) sin(x) + 201132771840cos(y) sin(x) --R + --R 14 9 16 8 --R - 574665062400cos(y) sin(x) + 1292996390400cos(y) sin(x) --R + --R 18 7 20 6 --R - 2298660249600cos(y) sin(x) + 3218124349440cos(y) sin(x) --R + --R 22 5 24 4 --R - 3510681108480cos(y) sin(x) + 2925567590400cos(y) sin(x) --R + --R 26 3 28 2 --R - 1800349286400cos(y) sin(x) + 771578265600cos(y) sin(x) --R + --R 30 32 --R - 205754204160cos(y) sin(x) + 25719275520cos(y) --R * --R 12 --R tan(z) --R + --R 17 2 16 4 15 --R 30780sin(x) - 1046520cos(y) sin(x) + 16744320cos(y) sin(x) --R + --R 6 14 8 13 --R - 167443200cos(y) sin(x) + 1172102400cos(y) sin(x) --R + --R 10 12 12 11 --R - 6094932480cos(y) sin(x) + 24379729920cos(y) sin(x) --R + --R 14 10 16 9 --R - 76622008320cos(y) sin(x) + 191555020800cos(y) sin(x) --R + --R 18 8 20 7 --R - 383110041600cos(y) sin(x) + 612976066560cos(y) sin(x) --R + --R 22 6 24 5 --R - 780151357440cos(y) sin(x) + 780151357440cos(y) sin(x) --R + --R 26 4 28 3 --R - 600116428800cos(y) sin(x) + 342923673600cos(y) sin(x) --R + --R 30 2 32 --R - 137169469440cos(y) sin(x) + 34292367360cos(y) sin(x) --R + --R 34 --R - 4034396160cos(y) --R * --R 9 --R tan(z) --R + --R 18 2 17 4 16 --R 1710sin(x) - 61560cos(y) sin(x) + 1046520cos(y) sin(x) --R + --R 6 15 8 14 --R - 11162880cos(y) sin(x) + 83721600cos(y) sin(x) --R + --R 10 13 12 12 --R - 468840960cos(y) sin(x) + 2031644160cos(y) sin(x) --R + --R 14 11 16 10 --R - 6965637120cos(y) sin(x) + 19155502080cos(y) sin(x) --R + --R 18 9 20 8 --R - 42567782400cos(y) sin(x) + 76622008320cos(y) sin(x) --R + --R 22 7 24 6 --R - 111450193920cos(y) sin(x) + 130025226240cos(y) sin(x) --R + --R 26 5 28 4 --R - 120023285760cos(y) sin(x) + 85730918400cos(y) sin(x) --R + --R 30 3 32 2 --R - 45723156480cos(y) sin(x) + 17146183680cos(y) sin(x) --R + --R 34 36 --R - 4034396160cos(y) sin(x) + 448266240cos(y) --R * --R 6 --R tan(z) --R + --R 19 2 18 4 17 --R 60sin(x) - 2280cos(y) sin(x) + 41040cos(y) sin(x) --R + --R 6 16 8 15 --R - 465120cos(y) sin(x) + 3720960cos(y) sin(x) --R + --R 10 14 12 13 --R - 22325760cos(y) sin(x) + 104186880cos(y) sin(x) --R + --R 14 12 16 11 --R - 386979840cos(y) sin(x) + 1160939520cos(y) sin(x) --R + --R 18 10 20 9 --R - 2837852160cos(y) sin(x) + 5675704320cos(y) sin(x) --R + --R 22 8 24 7 --R - 9287516160cos(y) sin(x) + 12383354880cos(y) sin(x) --R + --R 26 6 28 5 --R - 13335920640cos(y) sin(x) + 11430789120cos(y) sin(x) --R + --R 30 4 32 3 --R - 7620526080cos(y) sin(x) + 3810263040cos(y) sin(x) --R + --R 34 2 36 38 --R - 1344798720cos(y) sin(x) + 298844160cos(y) sin(x) - 31457280cos(y) --R * --R 3 --R tan(z) --R + --R 20 2 19 4 18 6 17 --R sin(x) - 40cos(y) sin(x) + 760cos(y) sin(x) - 9120cos(y) sin(x) --R + --R 8 16 10 15 12 14 --R 77520cos(y) sin(x) - 496128cos(y) sin(x) + 2480640cos(y) sin(x) --R + --R 14 13 16 12 --R - 9922560cos(y) sin(x) + 32248320cos(y) sin(x) --R + --R 18 11 20 10 --R - 85995520cos(y) sin(x) + 189190144cos(y) sin(x) --R + --R 22 9 24 8 --R - 343982080cos(y) sin(x) + 515973120cos(y) sin(x) --R + --R 26 7 28 6 --R - 635043840cos(y) sin(x) + 635043840cos(y) sin(x) --R + --R 30 5 32 4 --R - 508035072cos(y) sin(x) + 317521920cos(y) sin(x) --R + --R 34 3 36 2 --R - 149422080cos(y) sin(x) + 49807360cos(y) sin(x) --R + --R 38 40 --R - 10485760cos(y) sin(x) + 1048576cos(y) --R Type: Expression(Integer) --E 47 --S 48 of 63 factor(%) --R --R --R (48) --R 60 2 57 --R 3486784401tan(z) + (23245229340sin(x) - 46490458680cos(y) )tan(z) --R + --R 2 2 4 --R (73609892910sin(x) - 294439571640cos(y) sin(x) + 294439571640cos(y) ) --R * --R 54 --R tan(z) --R + --R 3 2 2 --R 147219785820sin(x) - 883318714920cos(y) sin(x) --R + --R 4 6 --R 1766637429840cos(y) sin(x) - 1177758286560cos(y) --R * --R 51 --R tan(z) --R + --R 4 2 3 --R 208561363245sin(x) - 1668490905960cos(y) sin(x) --R + --R 4 2 6 --R 5005472717880cos(y) sin(x) - 6673963623840cos(y) sin(x) --R + --R 8 --R 3336981811920cos(y) --R * --R 48 --R tan(z) --R + --R 5 2 4 --R 222465454128sin(x) - 2224654541280cos(y) sin(x) --R + --R 4 3 6 2 --R 8898618165120cos(y) sin(x) - 17797236330240cos(y) sin(x) --R + --R 8 10 --R 17797236330240cos(y) sin(x) - 7118894532096cos(y) --R * --R 45 --R tan(z) --R + --R 6 2 5 --R 185387878440sin(x) - 2224654541280cos(y) sin(x) --R + --R 4 4 6 3 --R 11123272706400cos(y) sin(x) - 29662060550400cos(y) sin(x) --R + --R 8 2 10 --R 44493090825600cos(y) sin(x) - 35594472660480cos(y) sin(x) --R + --R 12 --R 11864824220160cos(y) --R * --R 42 --R tan(z) --R + --R 7 2 6 --R 123591918960sin(x) - 1730286865440cos(y) sin(x) --R + --R 4 5 6 4 --R 10381721192640cos(y) sin(x) - 34605737308800cos(y) sin(x) --R + --R 8 3 10 2 --R 69211474617600cos(y) sin(x) - 83053769541120cos(y) sin(x) --R + --R 12 14 --R 55369179694080cos(y) sin(x) - 15819765626880cos(y) --R * --R 39 --R tan(z) --R + --R 8 2 7 --R 66945622770sin(x) - 1071129964320cos(y) sin(x) --R + --R 4 6 6 5 --R 7497909750240cos(y) sin(x) - 29991639000960cos(y) sin(x) --R + --R 8 4 10 3 --R 74979097502400cos(y) sin(x) - 119966556003840cos(y) sin(x) --R + --R 12 2 14 --R 119966556003840cos(y) sin(x) - 68552317716480cos(y) sin(x) --R + --R 16 --R 17138079429120cos(y) --R * --R 36 --R tan(z) --R + --R 9 2 8 --R 29753610120sin(x) - 535564982160cos(y) sin(x) --R + --R 4 7 6 6 --R 4284519857280cos(y) sin(x) - 19994426000640cos(y) sin(x) --R + --R 8 5 10 4 --R 59983278001920cos(y) sin(x) - 119966556003840cos(y) sin(x) --R + --R 12 3 14 2 --R 159955408005120cos(y) sin(x) - 137104635432960cos(y) sin(x) --R + --R 16 18 --R 68552317716480cos(y) sin(x) - 15233848381440cos(y) --R * --R 33 --R tan(z) --R + --R 10 2 9 --R 10909657044sin(x) - 218193140880cos(y) sin(x) --R + --R 4 8 6 7 --R 1963738267920cos(y) sin(x) - 10473270762240cos(y) sin(x) --R + --R 8 6 10 5 --R 36656447667840cos(y) sin(x) - 87975474402816cos(y) sin(x) --R + --R 12 4 14 3 --R 146625790671360cos(y) sin(x) - 167572332195840cos(y) sin(x) --R + --R 16 2 18 --R 125679249146880cos(y) sin(x) - 55857444065280cos(y) sin(x) --R + --R 20 --R 11171488813056cos(y) --R * --R 30 --R tan(z) --R + --R 11 2 10 --R 3305956680sin(x) - 72731046960cos(y) sin(x) --R + --R 4 9 6 8 --R 727310469600cos(y) sin(x) - 4363862817600cos(y) sin(x) --R + --R 8 7 10 6 --R 17455451270400cos(y) sin(x) - 48875263557120cos(y) sin(x) --R + --R 12 5 14 4 --R 97750527114240cos(y) sin(x) - 139643610163200cos(y) sin(x) --R + --R 16 3 18 2 --R 139643610163200cos(y) sin(x) - 93095740108800cos(y) sin(x) --R + --R 20 22 --R 37238296043520cos(y) sin(x) - 6770599280640cos(y) --R * --R 27 --R tan(z) --R + --R 12 2 11 --R 826489170sin(x) - 19835740080cos(y) sin(x) --R + --R 4 10 6 9 --R 218193140880cos(y) sin(x) - 1454620939200cos(y) sin(x) --R + --R 8 8 10 7 --R 6545794226400cos(y) sin(x) - 20946541524480cos(y) sin(x) --R + --R 12 6 14 5 --R 48875263557120cos(y) sin(x) - 83786166097920cos(y) sin(x) --R + --R 16 4 18 3 --R 104732707622400cos(y) sin(x) - 93095740108800cos(y) sin(x) --R + --R 20 2 22 --R 55857444065280cos(y) sin(x) - 20311797841920cos(y) sin(x) --R + --R 24 --R 3385299640320cos(y) --R * --R 24 --R tan(z) --R + --R 13 2 12 --R 169536240sin(x) - 4407942240cos(y) sin(x) --R + --R 4 11 6 10 --R 52895306880cos(y) sin(x) - 387898917120cos(y) sin(x) --R + --R 8 9 10 8 --R 1939494585600cos(y) sin(x) - 6982180508160cos(y) sin(x) --R + --R 12 7 14 6 --R 18619148021760cos(y) sin(x) - 37238296043520cos(y) sin(x) --R + --R 16 5 18 4 --R 55857444065280cos(y) sin(x) - 62063826739200cos(y) sin(x) --R + --R 20 3 22 2 --R 49651061391360cos(y) sin(x) - 27082397122560cos(y) sin(x) --R + --R 24 26 --R 9027465707520cos(y) sin(x) - 1388840878080cos(y) --R * --R 21 --R tan(z) --R + --R 14 2 13 --R 28256040sin(x) - 791169120cos(y) sin(x) --R + --R 4 12 6 11 --R 10285198560cos(y) sin(x) - 82281588480cos(y) sin(x) --R + --R 8 10 10 9 --R 452548736640cos(y) sin(x) - 1810194946560cos(y) sin(x) --R + --R 12 8 14 7 --R 5430584839680cos(y) sin(x) - 12412765347840cos(y) sin(x) --R + --R 16 6 18 5 --R 21722339358720cos(y) sin(x) - 28963119144960cos(y) sin(x) --R + --R 20 4 22 3 --R 28963119144960cos(y) sin(x) - 21064086650880cos(y) sin(x) --R + --R 24 2 26 --R 10532043325440cos(y) sin(x) - 3240628715520cos(y) sin(x) --R + --R 28 --R 462946959360cos(y) --R * --R 18 --R tan(z) --R + --R 15 2 14 4 13 --R 3767472sin(x) - 113024160cos(y) sin(x) + 1582338240cos(y) sin(x) --R + --R 6 12 8 11 --R - 13713598080cos(y) sin(x) + 82281588480cos(y) sin(x) --R + --R 10 10 12 9 --R - 362038989312cos(y) sin(x) + 1206796631040cos(y) sin(x) --R + --R 14 8 16 7 --R - 3103191336960cos(y) sin(x) + 6206382673920cos(y) sin(x) --R + --R 18 6 20 5 --R - 9654373048320cos(y) sin(x) + 11585247657984cos(y) sin(x) --R + --R 22 4 24 3 --R - 10532043325440cos(y) sin(x) + 7021362216960cos(y) sin(x) --R + --R 26 2 28 --R - 3240628715520cos(y) sin(x) + 925893918720cos(y) sin(x) --R + --R 30 --R - 123452522496cos(y) --R * --R 15 --R tan(z) --R + --R 16 2 15 4 14 --R 392445sin(x) - 12558240cos(y) sin(x) + 188373600cos(y) sin(x) --R + --R 6 13 8 12 --R - 1758153600cos(y) sin(x) + 11427998400cos(y) sin(x) --R + --R 10 11 12 10 --R - 54854392320cos(y) sin(x) + 201132771840cos(y) sin(x) --R + --R 14 9 16 8 --R - 574665062400cos(y) sin(x) + 1292996390400cos(y) sin(x) --R + --R 18 7 20 6 --R - 2298660249600cos(y) sin(x) + 3218124349440cos(y) sin(x) --R + --R 22 5 24 4 --R - 3510681108480cos(y) sin(x) + 2925567590400cos(y) sin(x) --R + --R 26 3 28 2 --R - 1800349286400cos(y) sin(x) + 771578265600cos(y) sin(x) --R + --R 30 32 --R - 205754204160cos(y) sin(x) + 25719275520cos(y) --R * --R 12 --R tan(z) --R + --R 17 2 16 4 15 --R 30780sin(x) - 1046520cos(y) sin(x) + 16744320cos(y) sin(x) --R + --R 6 14 8 13 --R - 167443200cos(y) sin(x) + 1172102400cos(y) sin(x) --R + --R 10 12 12 11 --R - 6094932480cos(y) sin(x) + 24379729920cos(y) sin(x) --R + --R 14 10 16 9 --R - 76622008320cos(y) sin(x) + 191555020800cos(y) sin(x) --R + --R 18 8 20 7 --R - 383110041600cos(y) sin(x) + 612976066560cos(y) sin(x) --R + --R 22 6 24 5 --R - 780151357440cos(y) sin(x) + 780151357440cos(y) sin(x) --R + --R 26 4 28 3 --R - 600116428800cos(y) sin(x) + 342923673600cos(y) sin(x) --R + --R 30 2 32 --R - 137169469440cos(y) sin(x) + 34292367360cos(y) sin(x) --R + --R 34 --R - 4034396160cos(y) --R * --R 9 --R tan(z) --R + --R 18 2 17 4 16 --R 1710sin(x) - 61560cos(y) sin(x) + 1046520cos(y) sin(x) --R + --R 6 15 8 14 --R - 11162880cos(y) sin(x) + 83721600cos(y) sin(x) --R + --R 10 13 12 12 --R - 468840960cos(y) sin(x) + 2031644160cos(y) sin(x) --R + --R 14 11 16 10 --R - 6965637120cos(y) sin(x) + 19155502080cos(y) sin(x) --R + --R 18 9 20 8 --R - 42567782400cos(y) sin(x) + 76622008320cos(y) sin(x) --R + --R 22 7 24 6 --R - 111450193920cos(y) sin(x) + 130025226240cos(y) sin(x) --R + --R 26 5 28 4 --R - 120023285760cos(y) sin(x) + 85730918400cos(y) sin(x) --R + --R 30 3 32 2 --R - 45723156480cos(y) sin(x) + 17146183680cos(y) sin(x) --R + --R 34 36 --R - 4034396160cos(y) sin(x) + 448266240cos(y) --R * --R 6 --R tan(z) --R + --R 19 2 18 4 17 --R 60sin(x) - 2280cos(y) sin(x) + 41040cos(y) sin(x) --R + --R 6 16 8 15 --R - 465120cos(y) sin(x) + 3720960cos(y) sin(x) --R + --R 10 14 12 13 --R - 22325760cos(y) sin(x) + 104186880cos(y) sin(x) --R + --R 14 12 16 11 --R - 386979840cos(y) sin(x) + 1160939520cos(y) sin(x) --R + --R 18 10 20 9 --R - 2837852160cos(y) sin(x) + 5675704320cos(y) sin(x) --R + --R 22 8 24 7 --R - 9287516160cos(y) sin(x) + 12383354880cos(y) sin(x) --R + --R 26 6 28 5 --R - 13335920640cos(y) sin(x) + 11430789120cos(y) sin(x) --R + --R 30 4 32 3 --R - 7620526080cos(y) sin(x) + 3810263040cos(y) sin(x) --R + --R 34 2 36 38 --R - 1344798720cos(y) sin(x) + 298844160cos(y) sin(x) - 31457280cos(y) --R * --R 3 --R tan(z) --R + --R 20 2 19 4 18 6 17 --R sin(x) - 40cos(y) sin(x) + 760cos(y) sin(x) - 9120cos(y) sin(x) --R + --R 8 16 10 15 12 14 --R 77520cos(y) sin(x) - 496128cos(y) sin(x) + 2480640cos(y) sin(x) --R + --R 14 13 16 12 --R - 9922560cos(y) sin(x) + 32248320cos(y) sin(x) --R + --R 18 11 20 10 --R - 85995520cos(y) sin(x) + 189190144cos(y) sin(x) --R + --R 22 9 24 8 --R - 343982080cos(y) sin(x) + 515973120cos(y) sin(x) --R + --R 26 7 28 6 --R - 635043840cos(y) sin(x) + 635043840cos(y) sin(x) --R + --R 30 5 32 4 --R - 508035072cos(y) sin(x) + 317521920cos(y) sin(x) --R + --R 34 3 36 2 --R - 149422080cos(y) sin(x) + 49807360cos(y) sin(x) --R + --R 38 40 --R - 10485760cos(y) sin(x) + 1048576cos(y) --R Type: Factored(Expression(Integer)) --E 48 \end{chunk} expand$[(1 - c^2)^5 (1 - s^2)^5 (c^2 + s^2)^{10}] => c^{10} s^{10}$ when $c^2 + s^2 = 1$ [modification of a problem due to Richard Liska] \begin{chunk}{*} --S 49 of 63 expand((1 - c**2)**5 * (1 - s**2)**5 * (c**2 + s**2)**10) --R --R --R (49) --R 10 8 6 4 2 30 --R (c - 5c + 10c - 10c + 5c - 1)s --R + --R 12 10 8 6 4 2 28 --R (10c - 55c + 125c - 150c + 100c - 35c + 5)s --R + --R 14 12 10 8 6 4 2 26 --R (45c - 275c + 710c - 1000c + 825c - 395c + 100c - 10)s --R + --R 16 14 12 10 8 6 4 2 --R 120c - 825c + 2425c - 3960c + 3900c - 2345c + 825c - 150c --R + --R 10 --R * --R 24 --R s --R + --R 18 16 14 12 10 8 6 --R 210c - 1650c + 5550c - 10450c + 12055c - 8735c + 3900c --R + --R 4 2 --R - 1000c + 125c - 5 --R * --R 22 --R s --R + --R 20 18 16 14 12 10 8 --R 252c - 2310c + 8970c - 19470c + 26060c - 22253c + 12055c --R + --R 6 4 2 --R - 3960c + 710c - 55c + 1 --R * --R 20 --R s --R + --R 22 20 18 16 14 12 10 --R 210c - 2310c + 10500c - 26400c + 40875c - 40645c + 26060c --R + --R 8 6 4 2 --R - 10450c + 2425c - 275c + 10c --R * --R 18 --R s --R + --R 24 22 20 18 16 14 12 --R 120c - 1650c + 8970c - 26400c + 47400c - 54615c + 40875c --R + --R 10 8 6 4 --R - 19470c + 5550c - 825c + 45c --R * --R 16 --R s --R + --R 26 24 22 20 18 16 14 --R 45c - 825c + 5550c - 19470c + 40875c - 54615c + 47400c --R + --R 12 10 8 6 --R - 26400c + 8970c - 1650c + 120c --R * --R 14 --R s --R + --R 28 26 24 22 20 18 16 --R 10c - 275c + 2425c - 10450c + 26060c - 40645c + 40875c --R + --R 14 12 10 8 --R - 26400c + 10500c - 2310c + 210c --R * --R 12 --R s --R + --R 30 28 26 24 22 20 18 --R c - 55c + 710c - 3960c + 12055c - 22253c + 26060c --R + --R 16 14 12 10 --R - 19470c + 8970c - 2310c + 252c --R * --R 10 --R s --R + --R 30 28 26 24 22 20 18 --R - 5c + 125c - 1000c + 3900c - 8735c + 12055c - 10450c --R + --R 16 14 12 --R 5550c - 1650c + 210c --R * --R 8 --R s --R + --R 30 28 26 24 22 20 18 --R 10c - 150c + 825c - 2345c + 3900c - 3960c + 2425c --R + --R 16 14 --R - 825c + 120c --R * --R 6 --R s --R + --R 30 28 26 24 22 20 18 16 4 --R (- 10c + 100c - 395c + 825c - 1000c + 710c - 275c + 45c )s --R + --R 30 28 26 24 22 20 18 2 30 28 --R (5c - 35c + 100c - 150c + 125c - 55c + 10c )s - c + 5c --R + --R 26 24 22 20 --R - 10c + 10c - 5c + c --R Type: Polynomial(Integer) --E 49 --S 50 of 63 groebner([%, c**2 + s**2 - 1]) --R --R --R 2 2 20 18 16 14 12 10 --R (50) [s + c - 1,c - 5c + 10c - 10c + 5c - c ] --R Type: List(Polynomial(Integer)) --E 50 --S 51 of 63 map(factor, %) --R --R --R 2 2 5 10 5 --R (51) [s + c - 1,(c - 1) c (c + 1) ] --R Type: List(Factored(Polynomial(Integer))) --E 51 \end{chunk} $=> (x + y) (x - y) {\textrm\ mod\ } 3$ \begin{chunk}{*} --S 52 of 63 factor(4*x**2 - 21*x*y + 20*y**2 :: Polynomial(PrimeField(3))) --R --R There are 22 exposed and 18 unexposed library operations named ** --R having 2 argument(s) but none was determined to be applicable. --R Use HyperDoc Browse, or issue --R )display op ** --R to learn more about the available operations. Perhaps --R package-calling the operation or using coercions on the arguments --R will allow you to apply the operation. --R --R Cannot find a definition or applicable library operation named ** --R with argument type(s) --R Variable(y) --R Polynomial(PrimeField(3)) --R --R Perhaps you should use "@" to indicate the required return type, --R or "$" to specify which version of the function you need. --E 52 \end{chunk} $=> 1/4 (x + y) (2 x + y [-1 + i sqrt(3)]) (2 x + y [-1 - i sqrt(3)])$ \begin{chunk}{*} --S 53 of 63 factor(x**3 + y**3, [rootOf(isqrt3**2 + 3)]) --R --R --R - isqrt3 - 1 isqrt3 - 1 --R (52) (y + ------------ x)(y + x)(y + ---------- x) --R 2 2 --R Type: Factored(Polynomial(AlgebraicNumber)) --E 53 \end{chunk} Partial fraction decomposition $=> 3/(x + 2) - 2/(x + 1) + 2/(x + 1)^2$ \begin{chunk}{*} --S 54 of 63 (x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2) --R --R --R 2 --R x + 2x + 3 --R (53) ----------------- --R 3 2 --R x + 4x + 5x + 2 --R Type: Fraction(Polynomial(Integer)) --E 54 --S 55 of 63 fullPartialFraction( _ % :: Fraction UnivariatePolynomial(x, Fraction Integer)) --R --R --R 2 2 3 --R (54) - ----- + -------- + ----- --R x + 1 2 x + 2 --R (x + 1) --RType: FullPartialFractionExpansion(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer))) --E 55 \end{chunk} Noncommutative algebra: note that $(A B C)^{(-1)} = C^{(-1)} B^{(-1)} A^{(-1)}$ $=> A B C A C B - C^{(-1)} B^{(-1)} C B$ \begin{chunk}{*} --S 56 of 63 A : SquareMatrix(2, Integer) --R --R Type: Void --E 56 --S 57 of 63 B : SquareMatrix(2, Integer) --R --R Type: Void --E 57 --S 58 of 63 C : SquareMatrix(2, Integer) --R --R Type: Void --E 58 --S 59 of 63 (A*B*C - (A*B*C)**(-1)) * A*C*B --R --R --R A is declared as being in SquareMatrix(2,Integer) but has not been --R given a value. --E 59 \end{chunk} Jacobi's identity: $[A, B, C] + [B, C, A] + [C, A, B] = 0$ where $[A, B, C] = [A, [B, C]]$ and $[A, B] = A B - B A$ is the commutator of $A$ and $B$ \begin{chunk}{*} --S 60 of 63 comm2(A, B) == A * B - B * A --R --R Type: Void --E 60 --S 61 of 63 comm3(A, B, C) == comm2(A, comm2(B, C)) --R --R Type: Void --E 61 --S 62 of 63 comm2(A, B) --R --R --R A is declared as being in SquareMatrix(2,Integer) but has not been --R given a value. --E 62 --S 63 of 63 comm3(A, B, C) + comm3(B, C, A) + comm3(C, A, B) --R --R --R A is declared as being in SquareMatrix(2,Integer) but has not been --R given a value. --E 63 )spool )lisp (bye) \end{chunk} \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}