Lösung 2.1:4b

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When the expression
When the expression
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<math>\left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right)</math>
 
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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.
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<math>\begin{align}
+
{{Displayed math||<math>\begin{align}
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& \left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right) \\
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&(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})\\[3pt]
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& =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) \\
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& +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} \\
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& +x^{3}\centerdot 2+x^{3}\centerdot \left( -x \right)+x^{3}\centerdot x^{2}+x^{3}\centerdot x^{4} \\
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&\qquad\qquad\quad{}+x^{3}\cdot 2+x^{3}\cdot (-x)+x^{3}\cdot x^{2}+x^{3}\cdot x^{4}\,\textrm{.}
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\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
+
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<math>x^{1}</math>
+
-
-term. In this case, we have two such pairs:
+
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<math>1</math>
+
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multiplied by -
+
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<math>x</math>
+
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and
+
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<math>x</math>
+
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multiplied by
+
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<math>2</math>
+
-
,
+
 +
so that the coefficient in front of ''x'' is <math>-1+2=1\,</math>.
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<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
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& \left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right)=...+1\centerdot \left( -x \right)+x\centerdot 2+... \\
+
-
& \\
+
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\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>.
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<math>x</math>
+
-
is
+
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<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>
+
-
.
+

Version vom 10:15, 23. Sep. 2008

When the expression

Vorlage:Displayed math

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

Vorlage:Displayed 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,

Vorlage:Displayed math

so that the coefficient in front of x is \displaystyle -1+2=1\,.

We obtain the coefficient in front of x² by finding those combinations of a term from each bracket which give an x²-term; these are

Vorlage:Displayed math

The coefficient in front of x² is \displaystyle 1-1+2=2\,.

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_2.1:4b“