Solution 2.1:4b - Revision history

Tek at 10:15, 23 September 2008

2008-09-23T10:15:48Z

←Older revision		Revision as of 10:15, 23 September 2008
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	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 at 14:24, 15 September 2008

2008-09-15T14:24:36Z

←Older revision		Revision as of 14:24, 15 September 2008
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-	{{~~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

2008-09-09T08:37:03Z

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

←Older revision

Revision as of 08:37, 9 September 2008

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

2008-08-21T06:35:39Z

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

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Lina at 09:58, 31 March 2008

2008-03-31T09:58:51Z

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