Solution 2.1:4b
From Förberedande kurs i matematik 1
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| - | {{ | + | When the expression |
| - | < | + | |
| - | {{ | + | {{Displayed math||<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})</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. | ||
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| + | {{Displayed math||<math>\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}</math>}} | ||
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| + | 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, | ||
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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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| + | 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 | ||
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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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| + | The coefficient in front of ''x''² is <math>1-1+2=2\,</math>. | ||
Current revision
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\,.
