4818.sagews

factor(x**2+2*x+2)

\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} + 2 \, x + 2

solve(x**2+2*x+2==0,x)

\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = \left(-i - 1\right), x = \left(i - 1\right)\right]

factor(x**2+2*x+1)

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x + 1\right)}^{2}

factor(x**2+2*x)

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x + 2\right)} x

factor(x**2+2*x-1)

\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} + 2 \, x - 1

solve(x**2+2*x-1==0,x)

\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = -\sqrt{2} - 1, x = \sqrt{2} - 1\right]

factor(10*x**2+11*x-6)

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(2 \, x + 3\right)} {\left(5 \, x - 2\right)}

factor(x**2-4)

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x - 2\right)} {\left(x + 2\right)}

factor(x**3-1)

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x - 1\right)} {\left(x^{2} + x + 1\right)}

factor(x**3+1)

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x + 1\right)} {\left(x^{2} - x + 1\right)}

factor(27*x**3-64)

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(3 \, x - 4\right)} {\left(9 \, x^{2} + 12 \, x + 16\right)}

var('y')
factor(125*x**3+8*y**3)

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(5 \, x + 2 \, y\right)} {\left(25 \, x^{2} - 10 \, x y + 4 \, y^{2}\right)}

factor(x**3+y**3)

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x + y\right)} {\left(x^{2} - x y + y^{2}\right)}

solve(x**2+2*x+2==0,x)

\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = \left(-i - 1\right), x = \left(i - 1\right)\right]

solve(x**2+2*x+1==0,x)

\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = \left(-1\right)\right]

solve(x**2+2*x==0,x)

\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = \left(-2\right), x = 0\right]

plot(x**2+2*x+2,(-2,2))

plot((x**2+2*x+2,x**2+2*x+1),(-2,2))

plot((x**2+2*x+2,x**2+2*x+1,x**2+2*x,x**2+2*x-1),(x,-2,2))

-1-sqrt(2)

\newcommand{\Bold}[1]{\mathbf{#1}}-\sqrt{2} - 1

n(-1-sqrt(2))

\newcommand{\Bold}[1]{\mathbf{#1}}-2.41421356237309

n(-1-sqrt(2),digits=1000)

\newcommand{\Bold}[1]{\mathbf{#1}}-2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912297024924836055850737212644121497099935831413222665927505592755799950501152782060571470109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989687253396546331808829640620615258352395054745750287759961729835575220337531857011354374603408498847160386899970699004815030544027790316454247823068492936918621580578463111596668713013015618568987237235288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162075847492265722600208558446652145839889394437092659180031138824646815708263010059485870400318648034219489727829064104507263688131373985525611732204024509122770022694112757362728049573810896750401836986836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112024944134172853147810580360337107730918286931471017111168391658172688941975871658215212822951848847

n(-1+sqrt(2),digits=1000)

\newcommand{\Bold}[1]{\mathbf{#1}}0.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727350138462309122970249248360558507372126441214970999358314132226659275055927557999505011527820605714701095599716059702745345968620147285174186408891986095523292304843087143214508397626036279952514079896872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471603868999706990048150305440277903164542478230684929369186215805784631115966687130130156185689872372352885092648612494977154218334204285686060146824720771435854874155657069677653720226485447015858801620758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342194897278290641045072636881313739855256117322040245091227700226941127573627280495738108967504018369868368450725799364729060762996941380475654823728997180326802474420629269124859052181004459842150591120249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472

(x+1)**0

\newcommand{\Bold}[1]{\mathbf{#1}}1

(x+1)**1

\newcommand{\Bold}[1]{\mathbf{#1}}x + 1

(x+1)**2

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x + 1\right)}^{2}

expand((x+1)**2)

\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} + 2 \, x + 1

expand((x+1)**3)

\newcommand{\Bold}[1]{\mathbf{#1}}x^{3} + 3 \, x^{2} + 3 \, x + 1

for i in range(0,4):
 show(expand((x+1)**i))

\newcommand{\Bold}[1]{\mathbf{#1}}1

\newcommand{\Bold}[1]{\mathbf{#1}}x + 1

\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} + 2 \, x + 1

\newcommand{\Bold}[1]{\mathbf{#1}}x^{3} + 3 \, x^{2} + 3 \, x + 1

for i in range(0,4):
 print(expand((x+1)**i))

1 x + 1 x^2 + 2*x + 1 x^3 + 3*x^2 + 3*x + 1

for i in range(0,4):
 (expand((x+1)**i))

\newcommand{\Bold}[1]{\mathbf{#1}}1

\newcommand{\Bold}[1]{\mathbf{#1}}x + 1

\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} + 2 \, x + 1

\newcommand{\Bold}[1]{\mathbf{#1}}x^{3} + 3 \, x^{2} + 3 \, x + 1

f(x)=expand((x+1)**6)
f(x)

\newcommand{\Bold}[1]{\mathbf{#1}}x^{6} + 6 \, x^{5} + 15 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} + 6 \, x + 1

f(-1)

\newcommand{\Bold}[1]{\mathbf{#1}}0

f(x)/(x+1)

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x^{6} + 6 \, x^{5} + 15 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} + 6 \, x + 1}{x + 1}

expand(f(x)/(x+1))

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x^{6}}{x + 1} + \frac{6 \, x^{5}}{x + 1} + \frac{15 \, x^{4}}{x + 1} + \frac{20 \, x^{3}}{x + 1} + \frac{15 \, x^{2}}{x + 1} + \frac{6 \, x}{x + 1} + \frac{1}{x + 1}

factor(f(x))/(x+1)

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x + 1\right)}^{5}

expand(factor(f(x))/(x+1))

\newcommand{\Bold}[1]{\mathbf{#1}}x^{5} + 5 \, x^{4} + 10 \, x^{3} + 10 \, x^{2} + 5 \, x + 1

f(x).diff()

\newcommand{\Bold}[1]{\mathbf{#1}}6 \, x^{5} + 30 \, x^{4} + 60 \, x^{3} + 60 \, x^{2} + 30 \, x + 6

var('t')
f(x)=expand((x+1)**6)
f(t)

\newcommand{\Bold}[1]{\mathbf{#1}}t^{6} + 6 \, t^{5} + 15 \, t^{4} + 20 \, t^{3} + 15 \, t^{2} + 6 \, t + 1

f(t).diff()

\newcommand{\Bold}[1]{\mathbf{#1}}6 \, t^{5} + 30 \, t^{4} + 60 \, t^{3} + 60 \, t^{2} + 30 \, t + 6

f(t).diff().diff()

\newcommand{\Bold}[1]{\mathbf{#1}}30 \, t^{4} + 120 \, t^{3} + 180 \, t^{2} + 120 \, t + 30

g(x)=f(x).diff(2);g(x)

\newcommand{\Bold}[1]{\mathbf{#1}}30 \, x^{4} + 120 \, x^{3} + 180 \, x^{2} + 120 \, x + 30

g(x)=f(x).diff(2);g(1)

\newcommand{\Bold}[1]{\mathbf{#1}}480

f(x)

\newcommand{\Bold}[1]{\mathbf{#1}}x^{6} + 6 \, x^{5} + 15 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} + 6 \, x + 1

f(x).integrate(x)

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{7} \, x^{7} + x^{6} + 3 \, x^{5} + 5 \, x^{4} + 5 \, x^{3} + 3 \, x^{2} + x

f(x).integrate(x).integrate(x)

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{56} \, x^{8} + \frac{1}{7} \, x^{7} + \frac{1}{2} \, x^{6} + x^{5} + \frac{5}{4} \, x^{4} + x^{3} + \frac{1}{2} \, x^{2}

f(x).integrate(x,0,1)

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{127}{7}

h(x)=(x+1)/(x**2+x-2);h(x)

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x + 1}{x^{2} + x - 2}

h(x)=(x+1)/(x**2+x-2)
factor(h(x))

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x + 1}{{\left(x - 1\right)} {\left(x + 2\right)}}

h(x).integrate(x)

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2}{3} \, \log\left(x - 1\right) + \frac{1}{3} \, \log\left(x + 2\right)

f(x)=x**2*e**(x)
f(x)

\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} e^{x}

f(x).integrate(x)

\newcommand{\Bold}[1]{\mathbf{#1}}{\left(x^{2} - 2 \, x + 2\right)} e^{x}

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_14.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("bGltaXQoKHjLhjIrMSkvKDIreCszKnjLhjIpLHg9aW5maW5pdHkp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmp7ICx_l/___code___.py", line 3 limit((xˆ_sage_const_2 +_sage_const_1 )/(_sage_const_2 +x+_sage_const_3 *xˆ_sage_const_2 ),x=infinity) ^ SyntaxError: invalid syntax