Answer: Option (C) is the correct answer.
Step-by-step explanation:
The given equation is as follows.
[tex]\frac{x}{x^{2}+3x+2} + \frac{3}{x+1}[/tex]
Taking L.C.M, the equation will become as follows.
[tex]\frac{x(x+1)+ 3(x^{2}+3x+2)}{(x^{2}+3x+2)(x+1)}[/tex] ........ (1)
Factorize the equation [tex]x^{2}+3x+2[/tex] in the denominator as follows.
[tex]x^{2}+3x+2[/tex]
= [tex]x^{2} + 2x + x + 2 [/tex]
= x(x+2) + 1(x + 2)
= (x+1)(x+2) ........ (2)
Put the factors in equation (2) in to equation (1), then the equation will become as follows.
[tex]\frac{x(x+1)+ 3(x^{2}+3x+2)}{(x^{2}+3x+2)(x+1)}[/tex]
= [tex]\frac{x^{2}+x +3x^{2}+9x+6)}{(x+1)(x+2)(x+1)}[/tex]
= [tex]\frac{4x^{2}+10x+6}{(x+1)^{2}(x+2)}[/tex]
Now, factorize the numerator as follows.
[tex]\frac{4x^{2}+10x+6}{(x+1)^{2}(x+2)}[/tex]
= [tex]\frac{4x^{2}+4x+6x+6}{(x+1)^{2}(x+2)}[/tex]
= [tex]\frac{4x(x+1) + 6(x+1)}{(x+1)^{2}(x+2)}[/tex]
= [tex]\frac{(4x+6)(x+1)}{(x+1)^{2}(x+2)}[/tex]
Cancelling (x+1) from both numerator and denominator. Then the equation will be written as follows.
[tex]\frac{(4x+6)(x+1)}{(x+1)^{2}(x+2)}[/tex]
= [tex]\frac{(4x+6)}{(x+1)(x+2)}[/tex]
Thus, we can conclude that the numerator of simplified sum is (4x+6).
The numerator of the simplified sum is 4x+6, the correct option is c.
Given
The given terms are;
[tex]\rm \dfrac{x}{x^2+3x+2 }+ \dfrac{3}{x+1}[/tex]
To find the sum first take LCM then simplify the equation and add the terms.
Therefore,
The sum of the equation is;
[tex]\rm =\dfrac{x}{x^2+3x+2 }+ \dfrac{3}{x+1}\\\\\rm =\dfrac{x }{(x+1)(x+2) } + \dfrac{3}{x+1}\\\\ = \dfrac{x+3(x+2)}{(x+1)(x+2)}\\\\= \dfrac{x+3x+6}{(x+1)(x+2)}\\\\= \dfrac{4x+6}{(x+1)(x+2)}\\\\[/tex]
Hence, the numerator of the simplified sum is 4x+6, the correct option is c.
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