A) \[9x\text{ }(x+1)\]
B) \[9x\left( 3x-1 \right)\]
C) \[18x\text{ }\left( 3x-1 \right)\]
D) \[18\text{ }x\text{ }(x+1)\]
Correct Answer: C
Solution :
Let \[P\left( x \right)\text{ }=\text{ }36\text{ }\left( 3{{x}^{4}}+\text{ }5{{x}^{3}}-\text{ }2{{x}^{2}} \right)\] \[=\text{ }36{{x}^{2}}\left( 3{{x}^{2}}+\text{ }5x\text{ }-\text{ }2 \right)\] \[=\text{ }36\,\,.\text{ }{{x}^{2}}\left\{ 3{{x}^{2}}+\text{ }6x\text{ }\text{ }x\text{ }-\text{ }2 \right\}\] \[=~~36.\text{ }{{x}^{2}}\left\{ 3x\text{ }\left( x\text{ }+\text{ }2 \right)\text{ }-\text{ }1\text{ }\left( x\text{ }+\text{ }2 \right) \right\}\] \[=2\times 2\times 3\times 3\times x\times x\times (x+2)(3x-1)\] \[Q\left( x \right)\text{ }=\text{ }9\left( 6{{x}^{3}}+\text{ }4{{x}^{2}}-\text{ }2x \right)\] \[=\text{ }9x\text{ }\left( 6{{x}^{2}}+\text{ }4x\text{ }-\text{ }2 \right)\] \[=~~18x\text{ }\left( 3{{x}^{2}}+\text{ }2x-1 \right)\] \[=~~18x\text{ }\{3{{x}^{2}}+\text{ }3x\text{ }-\text{ }x\text{ }-\text{ }1]\] \[=\text{ }332x.\text{ }\left( 3x-\text{ }1 \right)\text{ }\left( x+\text{ }1 \right)\] \[=\text{ }54\left( 27{{x}^{4}}-\text{ }x \right)\text{ }=\text{ }54x\text{ }\left( 27{{x}^{3}}-\text{ }1 \right)\] \[=\text{ }2\times 3\times 3\times 3\times x~\left( 3x-\text{ }1 \right)\] \[\left( 9{{x}^{2}}+\text{ }3x+\text{ }1 \right)\] So, HCF of \[\left[ P\left( x \right),\text{ }Q\left( x \right),\text{ }R\left( x \right) \right]\] \[=~~2~\times 3\times ~3\times ~x\text{ }\left( 3x-\text{ }1 \right)\] = \[\]
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