| If the LCM and HCF of two expressions are \[({{x}^{2}}+6x+8)(x+1)\] and \[(x+1)\]respectively and one of the expressions is \[{{x}^{2}}+3x+2,\]then find the other expression. [SSC (CGL) Mains 2014] |
A) \[{{x}^{2}}+5x+4\]
B) \[{{x}^{2}}-5x+4\]
C) \[{{x}^{2}}+4x+5\]
D) \[{{x}^{2}}-4x+5\]
Correct Answer: A
Solution :
| Given, \[\text{LCM}=({{x}^{2}}+6x+8)(x+1)\] |
| or \[(x+4)(x+2)(x+1)\] |
| and \[\text{HCF}=\text{(}x+1)\] |
| Ist expression \[={{x}^{2}}+3x+2\] or \[(x+1)(x+2)\] |
| As we know that, |
| Product of two expressions \[\text{LCM}\times \text{HCF}\] |
| \[\Rightarrow \]\[(x+1)(x+2)\times \text{2nd}\,\,\text{expression}\] |
| \[=(x+4)(x+2)(x+1)(x+1)\] |
| \[\therefore \]2nd expression \[=\frac{(x+4)(x+2)(x+1)(x+1)}{(x+1)(x+2)}\] |
| \[=(x+4)(x+1)\] |
| \[={{x}^{2}}+4x+x+4\] |
| \[={{x}^{2}}+5x+4\] |
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