Solution 2.1:4b

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m (Lösning 2.1:4b moved to Solution 2.1:4b: Robot: moved page)
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When the expression
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<center> [[Image:2_1_4b1(2).gif]] </center>
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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,
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<center> [[Image:2_1_4b-2(2).gif]] </center>
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every term in the first bracket is multiplied by every term in the second bracket, i.e.
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<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\centerdot 2+1\centerdot \left( -x \right)+1\centerdot x^{2}+1\centerdot x^{4}+x\centerdot 2+x\centerdot \left( -x \right) \\
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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} \\
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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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\end{align}</math>
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If we only want to know the coefficient in front of
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<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>
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-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>
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,
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<math>\begin{align}
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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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& \\
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\end{align}</math>
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so that the coefficient in front of
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<math>x</math>
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is
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<math>-1+2=1</math>
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We obtain the coefficient in front of
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<math>x^{2}</math>
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by finding those combinations of a term from each bracket
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which give an
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<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>
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 +
 
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The coefficient in front of
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<math>x^{2}</math>
 +
is
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<math>1-1+2</math>
 +
.

Revision as of 14:24, 15 September 2008

When the expression \displaystyle \left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right) is expanded out,

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


\displaystyle \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}


If we only want to know the coefficient in front of \displaystyle 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 \displaystyle x^{1} -term. In this case, we have two such pairs: \displaystyle 1 multiplied by - \displaystyle x and \displaystyle x multiplied by \displaystyle 2 ,


\displaystyle \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}


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


We obtain the coefficient in front of \displaystyle x^{2} by finding those combinations of a term from each bracket which give an \displaystyle x^{2} -term; these are


\displaystyle \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+...


The coefficient in front of \displaystyle x^{2} is \displaystyle 1-1+2 .

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