Lösung 2.1:4b

Aus Online Mathematik Brückenkurs 1

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K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
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When the expression
When the expression
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{{Displayed math||<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})</math>}}
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{{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.
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{{Displayed math||<math>\begin{align}
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{{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) \\
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multiplied by -''x'' and ''x'' multiplied by 2,
multiplied by -''x'' and ''x'' multiplied by 2,
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{{Displayed math||<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots</math>}}
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{{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>.
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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
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{{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>}}
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{{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>.

Version vom 08:23, 22. Okt. 2008

When the expression

\displaystyle (1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})

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

\displaystyle \begin{align}

&(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\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} \\ &\qquad\qquad\quad{}+x^{3}\cdot 2+x^{3}\cdot (-x)+x^{3}\cdot x^{2}+x^{3}\cdot x^{4}\,\textrm{.} \end{align}

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,

\displaystyle (1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4}) = \cdots + 1\cdot (-x) + x\cdot 2 + \cdots

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

\displaystyle (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

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“