Processing Math: Done
To print higher-resolution math symbols, click the
Hi-Res Fonts for Printing button on the jsMath control panel.

jsMath

Solution 2.1:4b

From Förberedande kurs i matematik 1

(Difference between revisions)
Jump to: navigation, search
m (Lösning 2.1:4b moved to Solution 2.1:4b: Robot: moved page)
Line 1: Line 1:
-
{{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>
 +
.

Revision as of 14:24, 15 September 2008

When the expression 1+x+x2+x32x+x2+x4  is expanded out,

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


1+x+x2+x32x+x2+x4=12+1x+1x2+1x4+x2+xx+xx2+xx4+x22+x2x+x2x2+x2x4+x32+x3x+x3x2+x3x4


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 x1 -term. In this case, we have two such pairs: 1 multiplied by - x and x multiplied by 2 ,


1+x+x2+x32x+x2+x4=+1x+x2+ 


so that the coefficient in front of x is 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


1+x+x2+x32x+x2+x4=+1x2+xx+x22+ 


The coefficient in front of x2 is 11+2 .