4348.sagews

Linear Factor Definition: In mathematics, the term linear function can refer to either of two different but related concepts: *a first degree polynomial function of one variable; *a map between two vector spaces that preserves vector addition and scalar multiplication.

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_3.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("TGluZWFyIEZhY3RvciBEZWZpbml0aW9uOiBJbiBtYXRoZW1hdGljcywgdGhlIHRlcm0gbGluZWFyIGZ1bmN0aW9uIGNhbiByZWZlciB0byBlaXRoZXIgb2YgdHdvIGRpZmZlcmVudCBidXQgcmVsYXRlZCBjb25jZXB0czogKmEgZmlyc3QgZGVncmVlIHBvbHlub21pYWwgZnVuY3Rpb24gb2Ygb25lIHZhcmlhYmxlOyAqYSBtYXAgYmV0d2VlbiB0d28gdmVjdG9yIHNwYWNlcyB0aGF0IHByZXNlcnZlcyB2ZWN0b3IgYWRkaXRpb24gYW5kIHNjYWxhciBtdWx0aXBsaWNhdGlvbi4="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpJbn2CN/___code___.py", line 2 Linear Factor Definition: In mathematics, the term linear function can refer to either of two different but related concepts: *a first degree polynomial function of one variable; *a map between two vector spaces that preserves vector addition and scalar multiplication. ^ SyntaxError: invalid syntax

Irreducible Quadratic Factor Definition: Quadratic polynomialhas no real roots. such a quadratic polynomial is called irreducible over the real numbers.

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_4.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("SXJyZWR1Y2libGUgUXVhZHJhdGljIEZhY3RvciBEZWZpbml0aW9uOiBRdWFkcmF0aWMgcG9seW5vbWlhbGhhcyBubyByZWFsIHJvb3RzLiBzdWNoIGEgcXVhZHJhdGljIHBvbHlub21pYWwgaXMgY2FsbGVkIGlycmVkdWNpYmxlIG92ZXIgdGhlIHJlYWwgbnVtYmVycy4="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmphUjJN6/___code___.py", line 2 Irreducible Quadratic Factor Definition: Quadratic polynomialhas no real roots. such a quadratic polynomial is called irreducible over the real numbers. ^ SyntaxError: invalid syntax

Fundamental Theorem of Algebra Definition: In mathematics, the fundamental theorem of algebra states that every non-constant single-variable  polynomial with complex coefficients has at least on complex root. Equivalently, the field of complex numbers is algebraically closed.

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_5.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("RnVuZGFtZW50YWwgVGhlb3JlbSBvZiBBbGdlYnJhIERlZmluaXRpb246IEluIG1hdGhlbWF0aWNzLCB0aGUgZnVuZGFtZW50YWwgdGhlb3JlbSBvZiBhbGdlYnJhIHN0YXRlcyB0aGF0IGV2ZXJ5IG5vbi1jb25zdGFudCBzaW5nbGUtdmFyaWFibGUgIHBvbHlub21pYWwgd2l0aCBjb21wbGV4IGNvZWZmaWNpZW50cyBoYXMgYXQgbGVhc3Qgb24gY29tcGxleCByb290LiBFcXVpdmFsZW50bHksIHRoZSBmaWVsZCBvZiBjb21wbGV4IG51bWJlcnMgaXMgYWxnZWJyYWljYWxseSBjbG9zZWQu"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmptvWDl0/___code___.py", line 2 Fundamental Theorem of Algebra Definition: In mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least on complex root. Equivalently, the field of complex numbers is algebraically closed. ^ SyntaxError: invalid syntax

y=x^3-x^2-6*x

factor(y)

(x - 3)*(x + 2)*x

y=(x^3)+1

factor(y)

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

y=x^4+4*x^3+8*x^2+8*x+4

factor(y)

(x^2 + 2*x + 2)^2

y=2*x^4-11*x^3+18*x^2-4*x-8

factor(y)

(x - 2)^3*(2*x + 1)

y=x^4+4*x^2+3

factor(y)

(x^2 + 1)*(x^2 + 3)

y=x^3-3*x^2

factor(y)

(x - 3)*x^2

f=(4*x^2-7*x-12)/(x^3-x^2-6*x)

f.partial_fraction(x)

1/5/(x - 3) + 9/5/(x + 2) + 2/x

integrate((4*x^2-7*x-12)/(x^3-x^2-6*x),x)

1/5*log(x - 3) + 9/5*log(x + 2) + 2*log(x)

f=(x^3)/(x^3+1)

f.partial_fraction(x)

1/3*(x - 2)/(x^2 - x + 1) - 1/3/(x + 1) + 1

integrate((x^3)/(x^3+1),x)

-1/3*sqrt(3)*arctan(1/3*(2*x - 1)*sqrt(3)) + x - 1/3*log(x + 1) + 1/6*log(x^2 - x + 1)

f=(x^3+2*x^2+3*x-2)/(x^4+4*x^3+8*x^2+8*x+4)

f.partial_fraction(x)

x/(x^2 + 2*x + 2) + (x - 2)/(x^2 + 2*x + 2)^2

integrate((x^3+2*x^2+3*x-2)/(x^4+4*x^3+8*x^2+8*x+4),x)

-1/2*(3*x + 4)/(x^2 + 2*x + 2) + 1/2*log(x^2 + 2*x + 2) - 5/2*arctan(x + 1)

f=(x^3-7*x+26*x-32)/(2*x^4-11*x^3+18*x^2-4*x-8)

f.partial_fraction(x)

-104/125/(x - 2) + 127/25/(x - 2)^2 + 14/5/(x - 2)^3 + 333/125/(2*x + 1)

integrate((x^3-7*x+26*x-32)/(2*x^4-11*x^3+18*x^2-4*x-8),x)

-1/25*(127*x - 219)/(x^2 - 4*x + 4) - 104/125*log(x - 2) + 333/250*log(2*x + 1)

f=(x^3+2*x)/(x^4+4*x^2+3)

f.partial_fraction(x)

1/2*x/(x^2 + 1) + 1/2*x/(x^2 + 3)

integrate((x^3+2*x)/(x^4+4*x^2+3),x)

1/4*log(x^4 + 4*x^2 + 3)

f=(x^4-5*x^3+6*x^2-18)/(x^3-3*x^2)

f.partial_fraction(x)

x - 2/(x - 3) + 2/x + 6/x^2 - 2

integrate((x^4-5*x^3+6*x^2-18)/(x^3-3*x^2),x)

1/2*x^2 - 2*x - 6/x - 2*log(x - 3) + 2*log(x)

y=30*x^5-13*x^4+50*x^3-286*x^2-299*x-70

factor(y)

(2*x + 1)*(3*x - 7)*(5*x + 2)*(x^2 + x + 5)

f=(4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70)

f.partial_fraction(x)

1/260015*(22098*x + 48935)/(x^2 + x + 5) - 668/323/(2*x + 1) - 9438/80155/(3*x - 7) + 24110/4879/(5*x + 2)

integrate((4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70),x)

3988/260015*sqrt(19)*arctan(1/19*(2*x + 1)*sqrt(19)) - 334/323*log(2*x + 1) - 3146/80155*log(3*x - 7) + 4822/4879*log(5*x + 2) + 11049/260015*log(x^2 + x + 5)