Lösung 2.1:4b - Versionsgeschichte

Moni um 14:33, 4. Aug. 2009

Moni — Tue, 04 Aug 2009 14:33:10 GMT

← Nächstältere Version		Version vom 14:33, 4. Aug. 2009
Zeile 16:		Zeile 16:
	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}		{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}

-	und also ist der Koeffizient von ''x'' <math>-1+2=1\,</math>.	+	und also ist der Koeffizient von ''x'' : <math>-1+2=1\,</math>.

	Den Koeffizienten von den <math>x^2</math>-Term finden wir, indem wir alle Terme, die multipliziert einen <math>x^2</math>-Term ergeben multiplizieren, also		Den Koeffizienten von den <math>x^2</math>-Term finden wir, indem wir alle Terme, die multipliziert einen <math>x^2</math>-Term ergeben multiplizieren, also

Markus.bez um 08:14, 9. Jun. 2009

Markus.bez — Tue, 09 Jun 2009 08:14:23 GMT

← Nächstältere Version		Version vom 08:14, 9. Jun. 2009
Zeile 3:		Zeile 3:
	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})</math>}}		{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})</math>}}

-	erweitern, wird ~~jemer~~ Term n der ersten Klammer mit jedem Term in der zweiten Klammer multipliziert, also	+	erweitern, wird jeder Term der ersten Klammer mit jedem Term der zweiten Klammer multipliziert, also

	{{Abgesetzte Formel\|\|<math>\begin{align}		{{Abgesetzte Formel\|\|<math>\begin{align}
Zeile 12:		Zeile 12:
	\end{align}</math>}}		\end{align}</math>}}

-	Falls wir wie in diesem Fall nur den ''x''-Koeffizienten wissen wollen, genügt es wenn wir die Terme die multipliziert einen ''x''-Term ergeben multiplizieren. In diesem Fall entspricht das 1 mal -''x'' und ''x'' mal 2,	+	Falls wir wie in diesem Fall nur den ''x''-Koeffizienten wissen wollen, genügt es wenn wir die Terme, die multipliziert einen ''x''-Term ergeben, multiplizieren. In diesem Fall entspricht das 1 mal -''x'' und ''x'' mal 2,

	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}		{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}
Zeile 18:		Zeile 18:
	und also ist der Koeffizient von ''x'' <math>-1+2=1\,</math>.		und also ist der Koeffizient von ''x'' <math>-1+2=1\,</math>.

-	Den Koeffizienten von den <math>x^2</math>-Term finden wir indem wir alle Terme die multipliziert einen <math>x^2</math>-Term ergeben multiplizieren, also	+	Den Koeffizienten von den <math>x^2</math>-Term finden wir, indem wir alle Terme, die multipliziert einen <math>x^2</math>-Term ergeben multiplizieren, also

	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot x^{2} + x\cdot(-x) + x^{2}\cdot 2 + \cdots</math>}}		{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot x^{2} + x\cdot(-x) + x^{2}\cdot 2 + \cdots</math>}}

	Der Koeffizient von <math>x^2</math> ist also <math>1-1+2=2\,</math>.		Der Koeffizient von <math>x^2</math> ist also <math>1-1+2=2\,</math>.

Martin um 19:23, 28. Feb. 2009

Martin — Sat, 28 Feb 2009 19:23:46 GMT

← Nächstältere Version		Version vom 19:23, 28. Feb. 2009
Zeile 12:		Zeile 12:
	\end{align}</math>}}		\end{align}</math>}}

-	Falls wir wie in diesem Fall nur den ''x''-Koeffizienten wissen wollen, genügt es wenn wir die Terme ~~multiplixieren,~~ die einen ''x''-Term ergeben. In diesem Fall entspricht das 1 mal -''x'' und ''x'' mal 2,	+	Falls wir wie in diesem Fall nur den ''x''-Koeffizienten wissen wollen, genügt es wenn wir die Terme die multipliziert einen ''x''-Term ergeben multiplizieren. In diesem Fall entspricht das 1 mal -''x'' und ''x'' mal 2,

	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}		{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}
Zeile 18:		Zeile 18:
	und also ist der Koeffizient von ''x'' <math>-1+2=1\,</math>.		und also ist der Koeffizient von ''x'' <math>-1+2=1\,</math>.

-	Den Koeffizienten von den <math>x^2</math>-Term finden wir indem wir alle Terme ~~multiplizieren~~ die einen <math>x^2</math>-Term ergeben, also	+	Den Koeffizienten von den <math>x^2</math>-Term finden wir indem wir alle Terme die multipliziert einen <math>x^2</math>-Term ergeben multiplizieren, also

	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot x^{2} + x\cdot(-x) + x^{2}\cdot 2 + \cdots</math>}}		{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot x^{2} + x\cdot(-x) + x^{2}\cdot 2 + \cdots</math>}}

	Der Koeffizient von <math>x^2</math> ist also <math>1-1+2=2\,</math>.		Der Koeffizient von <math>x^2</math> ist also <math>1-1+2=2\,</math>.

Martin um 19:22, 28. Feb. 2009

Martin — Sat, 28 Feb 2009 19:22:23 GMT

← Nächstältere Version		Version vom 19:22, 28. Feb. 2009
Zeile 1:		Zeile 1:
-	~~When the expression~~	+	Wenn wir den Ausdruck

	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})</math>}}		{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})</math>}}

-	~~is expanded out~~, ~~every term~~ in ~~the first bracket is multiplied by every term in the second bracket~~, ~~i.e.~~	+	erweitern, wird jemer Term n der ersten Klammer mit jedem Term in der zweiten Klammer multipliziert, also

	{{Abgesetzte Formel\|\|<math>\begin{align}		{{Abgesetzte Formel\|\|<math>\begin{align}
Zeile 12:		Zeile 12:
	\end{align}</math>}}		\end{align}</math>}}

-	~~If we only want to know the coefficient~~ in ~~front of~~ ''x'', ~~we do not need to carry out the complete expansion of the expression; it is sufficient to find those combinations of a term from the first bracket and a term from the second bracket which~~, ~~when multiplied, give an~~ ''x''-~~term~~. In ~~this case, we have two such pairs:~~ 1	+	Falls wir wie in diesem Fall nur den ''x''-Koeffizienten wissen wollen, genügt es wenn wir die Terme multiplixieren, die einen ''x''-Term ergeben. In diesem Fall entspricht das 1 mal -''x'' und ''x'' mal 2,
-	~~multiplied by~~ -''x'' ~~and~~ ''x'' ~~multiplied by~~ 2,	+

	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}		{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}

-	~~so that the coefficient in front of~~ ''x'' is <math>-1+2=1\,</math>.	+	und also ist der Koeffizient von ''x'' <math>-1+2=1\,</math>.

-	~~We obtain the coefficient in front of ''~~x~~''² by finding those combinations of a term from each bracket which give an ''~~x~~''²~~-~~term; these are~~	+	Den Koeffizienten von den <math>x^2</math>-Term finden wir indem wir alle Terme multiplizieren die einen <math>x^2</math>-Term ergeben, also

	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot x^{2} + x\cdot(-x) + x^{2}\cdot 2 + \cdots</math>}}		{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot x^{2} + x\cdot(-x) + x^{2}\cdot 2 + \cdots</math>}}

-	~~The coefficient in front of ''~~x~~''² is~~ <math>1-1+2=2\,</math>.	+	Der Koeffizient von <math>x^2</math> ist also <math>1-1+2=2\,</math>.

Tekbot: hat „Solution 2.1:4b“ nach „Lösung 2.1:4b“ verschoben: Robot: moved page

Tekbot — Wed, 22 Oct 2008 13:47:25 GMT

hat „Solution 2.1:4b“ nach „Lösung 2.1:4b“ verschoben: Robot: moved page

← Nächstältere Version

Version vom 13:47, 22. Okt. 2008

Tekbot: Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

Tekbot — Wed, 22 Oct 2008 08:23:38 GMT

Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

Tek um 10:15, 23. Sep. 2008

Tek — Tue, 23 Sep 2008 10:15:48 GMT

← Nächstältere Version		Version vom 10:15, 23. Sep. 2008
Zeile 1:		Zeile 1:
	When the expression		When the expression
-	<math>\left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right)</math>
-	is expanded out,

-	~~every term in the first bracket is multiplied by every term in the second bracket, i.e.~~	+	{{Displayed math\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})</math>}}

		+	is expanded out, every term in the first bracket is multiplied by every term in the second bracket, i.e.

-	<math>\begin{align}	+	{{Displayed math\|\|<math>\begin{align}
-	& ~~\left~~( 1+x+x^{2}+x^{3} ~~\right~~)~~\left~~( 2-x+x^{2}+x^{4} ~~\right~~) \\	+	&(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})\\[3pt]
-	& =1\~~centerdot~~ 2+1\~~centerdot \left~~( -x ~~\right~~)+1\~~centerdot~~ x^{2}+1\~~centerdot~~ x^{4}+x\~~centerdot~~ 2+x\~~centerdot \left~~( -x ~~\right~~) \\	+	&\qquad\quad{}=1\cdot 2+1\cdot (-x)+1\cdot x^{2}+1\cdot x^{4}+x\cdot 2+x\cdot (-x) \\
-	& +x\~~centerdot~~ x^{2}+x\~~centerdot~~ x^{4}+x^{2}\~~centerdot~~ 2+x^{2}\~~centerdot \left~~( -x ~~\right~~)+x^{2}\~~centerdot~~ x^{2}+x^{2}\~~centerdot~~ x^{4} \\	+	&\qquad\qquad\quad{}+x\cdot x^{2}+x\cdot x^{4}+x^{2}\cdot 2+x^{2}\cdot (-x)+x^{2}\cdot x^{2}+x^{2}\cdot x^{4} \\
-	& +x^{3}\~~centerdot~~ 2+x^{3}\~~centerdot \left~~( -x ~~\right~~)+x^{3}\~~centerdot~~ x^{2}+x^{3}\~~centerdot~~ x^{4} \\	+	&\qquad\qquad\quad{}+x^{3}\cdot 2+x^{3}\cdot (-x)+x^{3}\cdot x^{2}+x^{3}\cdot x^{4}\,\textrm{.}
-	\end{align}</math>	+	\end{align}</math>}}

		+	If we only want to know the coefficient in front of ''x'', we do not need to carry out the complete expansion of the expression; it is sufficient to find those combinations of a term from the first bracket and a term from the second bracket which, when multiplied, give an ''x''-term. In this case, we have two such pairs: 1
		+	multiplied by -''x'' and ''x'' multiplied by 2,

-	~~If we only want to know the coefficient in front of~~	+	{{Displayed math\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}
-	<math>x</math>, we do not need to carry out the complete expansion of the expression; it is sufficient to find those combinations of a term from the first bracket and a term from the second bracket which, when multiplied, give an	+
-	<math>x^{1}~~</math>~~	+
-	-~~term. In this case, we have two such pairs:~~	+
-	~~<math>~~1~~</math>~~	+
-	~~multiplied by~~ -	+
-	~~<math>~~x~~</math>~~	+
-	~~and~~	+
-	~~<math>~~x~~</math>~~	+
-	~~multiplied by~~	+
-	~~<math>~~2</math>	+
-	,	+

		+	so that the coefficient in front of ''x'' is <math>-1+2=1\,</math>.

-	~~<math>\begin{align}~~	+	We obtain the coefficient in front of ''x''² by finding those combinations of a term from each bracket which give an ''x''²-term; these are
-	~~& \left( 1+~~x+x~~^{2}+x^{3} \right)\left( 2~~-~~x+x^{2}+x^{4} \right)=...+1\centerdot \left( -x \right)+x\centerdot 2+... \\~~	+
-	~~& \\~~	+
-	~~\end{align}</math>~~	+

		+	{{Displayed math\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot x^{2} + x\cdot(-x) + x^{2}\cdot 2 + \cdots</math>}}

-	~~so that the~~ coefficient in front of	+	The coefficient in front of ''x''² is <math>1-1+2=2\,</math>.
-	~~<math>~~x~~</math>~~	+
-	is	+
-	<math>-1~~+2=1</math>~~	+
-		+
-		+
-	~~We obtain the coefficient in front of~~	+
-	~~<math>x^{2}</math>~~	+
-	~~by finding those combinations of a term from each bracket~~	+
-	~~which give an~~	+
-	~~<math>x^{2}</math>~~	+
-	-~~term; these are~~	+
-		+
-		+
-	~~<math>\left(~~ 1+~~x+x^{~~2~~}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right)~~=~~...+1\centerdot x^{~~2~~}+x~~\~~centerdot \left( -x \right)+x^{2}\centerdot 2+...</math>~~	+
-		+
-		+
-	~~The coefficient in front of~~	+
-	~~<math>x^{2}</math>~~	+
-	is	+
-	~~<math>1-1+2~~</math>	+
-	.	+

Ian um 14:24, 15. Sep. 2008

Ian — Mon, 15 Sep 2008 14:24:36 GMT

← Nächstältere Version		Version vom 14:24, 15. Sep. 2008
Zeile 1:		Zeile 1:
-	{{~~NAVCONTENT_START~~}}	+	When the expression
-	<~~center~~> ~~[[Image:2_1_4b1~~(2)~~.gif]] </center>~~	+	<math>\left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right)</math>
-	{{~~NAVCONTENT_STOP~~}}	+	is expanded out,
-	{{~~NAVCONTENT_START~~}}	+
-	<~~center~~> ~~[[Image~~:~~2_1_4b~~-2(2).~~gif]]~~ </~~center~~>	+	every term in the first bracket is multiplied by every term in the second bracket, i.e.
-	{{~~NAVCONTENT_STOP~~}}	+
		+
		+	<math>\begin{align}
		+	& \left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right) \\
		+	& =1\centerdot 2+1\centerdot \left( -x \right)+1\centerdot x^{2}+1\centerdot x^{4}+x\centerdot 2+x\centerdot \left( -x \right) \\
		+	& +x\centerdot x^{2}+x\centerdot x^{4}+x^{2}\centerdot 2+x^{2}\centerdot \left( -x \right)+x^{2}\centerdot x^{2}+x^{2}\centerdot x^{4} \\
		+	& +x^{3}\centerdot 2+x^{3}\centerdot \left( -x \right)+x^{3}\centerdot x^{2}+x^{3}\centerdot x^{4} \\
		+	\end{align}</math>
		+
		+
		+	If we only want to know the coefficient in front of
		+	<math>x</math>, we do not need to carry out the complete expansion of the expression; it is sufficient to find those combinations of a term from the first bracket and a term from the second bracket which, when multiplied, give an
		+	<math>x^{1}</math>
		+	-term. In this case, we have two such pairs:
		+	<math>1</math>
		+	multiplied by -
		+	<math>x</math>
		+	and
		+	<math>x</math>
		+	multiplied by
		+	<math>2</math>
		+	,
		+
		+
		+	<math>\begin{align}
		+	& \left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right)=...+1\centerdot \left( -x \right)+x\centerdot 2+... \\
		+	& \\
		+	\end{align}</math>
		+
		+
		+	so that the coefficient in front of
		+	<math>x</math>
		+	is
		+	<math>-1+2=1</math>
		+
		+
		+	We obtain the coefficient in front of
		+	<math>x^{2}</math>
		+	by finding those combinations of a term from each bracket
		+	which give an
		+	<math>x^{2}</math>
		+	-term; these are
		+
		+
		+	<math>\left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right)=...+1\centerdot x^{2}+x\centerdot \left( -x \right)+x^{2}\centerdot 2+...</math>
		+
		+
		+	The coefficient in front of
		+	<math>x^{2}</math>
		+	is
		+	<math>1-1+2</math>
		+	.

Tekbot: Lösning 2.1:4b moved to Solution 2.1:4b: Robot: moved page

Tekbot — Tue, 09 Sep 2008 08:37:03 GMT

Lösning 2.1:4b moved to Solution 2.1:4b: Robot: moved page

← Nächstältere Version

Version vom 08:37, 9. Sep. 2008

Tekbot: Robot: Automated text replacement (-[[Bild: +[[Image:)

Tekbot — Thu, 21 Aug 2008 06:35:39 GMT

Robot: Automated text replacement (-[[Bild: +[[Image:)

← Nächstältere Version		Version vom 06:35, 21. Aug. 2008
Zeile 1:		Zeile 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:2_1_4b1(2).gif]] </center>	+	<center> [[Image:2_1_4b1(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:2_1_4b-2(2).gif]] </center>	+	<center> [[Image:2_1_4b-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}

← Nächstältere Version		Version vom 08:23, 22. Okt. 2008
Zeile 1:		Zeile 1:
	When the expression		When the expression

-	{{~~Displayed math~~\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})</math>}}	+	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})</math>}}

	is expanded out, every term in the first bracket is multiplied by every term in the second bracket, i.e.		is expanded out, every term in the first bracket is multiplied by every term in the second bracket, i.e.

-	{{~~Displayed math~~\|\|<math>\begin{align}	+	{{Abgesetzte Formel\|\|<math>\begin{align}
	&(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})\\[3pt]		&(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})\\[3pt]
	&\qquad\quad{}=1\cdot 2+1\cdot (-x)+1\cdot x^{2}+1\cdot x^{4}+x\cdot 2+x\cdot (-x) \\		&\qquad\quad{}=1\cdot 2+1\cdot (-x)+1\cdot x^{2}+1\cdot x^{4}+x\cdot 2+x\cdot (-x) \\
Zeile 15:		Zeile 15:
	multiplied by -''x'' and ''x'' multiplied by 2,		multiplied by -''x'' and ''x'' multiplied by 2,

-	{{~~Displayed math~~\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}	+	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}

	so that the coefficient in front of ''x'' is <math>-1+2=1\,</math>.		so that the coefficient in front of ''x'' is <math>-1+2=1\,</math>.
Zeile 21:		Zeile 21:
	We obtain the coefficient in front of ''x''² by finding those combinations of a term from each bracket which give an ''x''²-term; these are		We obtain the coefficient in front of ''x''² by finding those combinations of a term from each bracket which give an ''x''²-term; these are

-	{{~~Displayed math~~\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot x^{2} + x\cdot(-x) + x^{2}\cdot 2 + \cdots</math>}}	+	{{Abgesetzte Formel\|\|<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot x^{2} + x\cdot(-x) + x^{2}\cdot 2 + \cdots</math>}}

	The coefficient in front of ''x''² is <math>1-1+2=2\,</math>.		The coefficient in front of ''x''² is <math>1-1+2=2\,</math>.