Solution 2.1:5c

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The fraction can be further simplified if it is possible to factorize and eliminate common factors
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<center> [[Image:2_1_5c.gif]] </center>
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from the numerator and denominator. Both numerator and denominator are already factorized to a certain extent, but we can go further with the numerator and break it up into linear factors by using the conjugate rule:
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<math>\begin{align}
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& 3x^{2}-12=3\left( x^{2}-4 \right)=3\left( x+2 \right)\left( x-2 \right) \\
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& \\
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& x^{2}-1=\left( x+1 \right)\left( x-1 \right) \\
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\end{align}</math>
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The whole expression is therefore equal to
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<math>\frac{3\left( x+2 \right)\left( x-2 \right)\left( x+1 \right)\left( x-1 \right)}{\left( x+1 \right)\left( x+2 \right)}=3\left( x-2 \right)\left( x-1 \right)</math>
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NOTE: One can of course expand out the expression to get
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<math>3x^{2}-9x+6</math>
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as the answer.

Revision as of 09:26, 16 September 2008

The fraction can be further simplified if it is possible to factorize and eliminate common factors from the numerator and denominator. Both numerator and denominator are already factorized to a certain extent, but we can go further with the numerator and break it up into linear factors by using the conjugate rule:


\displaystyle \begin{align} & 3x^{2}-12=3\left( x^{2}-4 \right)=3\left( x+2 \right)\left( x-2 \right) \\ & \\ & x^{2}-1=\left( x+1 \right)\left( x-1 \right) \\ \end{align}

The whole expression is therefore equal to


\displaystyle \frac{3\left( x+2 \right)\left( x-2 \right)\left( x+1 \right)\left( x-1 \right)}{\left( x+1 \right)\left( x+2 \right)}=3\left( x-2 \right)\left( x-1 \right)


NOTE: One can of course expand out the expression to get \displaystyle 3x^{2}-9x+6 as the answer.

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