From Förberedande kurs i matematik 1

Revision as of 08:37, 9 September 2008 (edit) Tekbot (Talk | contribs) m (Lösning 2.1:4b moved to Solution 2.1:4b: Robot: moved page) ← Previous diff		Current revision (10:15, 23 September 2008) (edit) (undo) Tek (Talk | contribs) m
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-	{{NAVCONTENT_START}}	+	When the expression
-	<center> [[Image:2_1_4b1(2).gif]] </center>	+
-	{{NAVCONTENT_STOP}}	+	{{Displayed math||<math>(1+x+x^{2}+x^{3})(2-x+x^{2}+x^{4})</math>}}
-	{{NAVCONTENT_START}}	+
-	<center> [[Image:2_1_4b-2(2).gif]] </center>	+	is expanded out, every term in the first bracket is multiplied by every term in the second bracket, i.e.
-	{{NAVCONTENT_STOP}}	+
		+	{{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>}}
		+
		+	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,
		+
		+	{{Displayed math||<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>.
		+
		+	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>}}
		+
		+	The coefficient in front of ''x''² is <math>1-1+2=2\,</math>.

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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\,.