Question:
Find the sum of the terms. x/x^2+3x+2 + 3/x+1
The numerator of the simplified sum is ...?
A) x + 3
B) 3x + 6
C) 4x + 6
D) 4x + 2.
Answer & Step-by-step explanation:
The given equation is as follows.
\frac{x}{x^{2}+3x+2} + \frac{3}{x+1}
Taking L.C.M, the equation will become as follows.
\frac{x(x+1)+ 3(x^{2}+3x+2)}{(x^{2}+3x+2)(x+1)} ........ (1)
Factorize the equation x^{2}+3x+2 in the denominator as follows.
x^{2}+3x+2
= x^{2} + 2x + x + 2
= 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.
\frac{x(x+1)+ 3(x^{2}+3x+2)}{(x^{2}+3x+2)(x+1)}
= \frac{x^{2}+x +3x^{2}+9x+6)}{(x+1)(x+2)(x+1)}
= \frac{4x^{2}+10x+6}{(x+1)^{2}(x+2)}
Now, factorize the numerator as follows.
\frac{4x^{2}+10x+6}{(x+1)^{2}(x+2)}
= \frac{4x^{2}+4x+6x+6}{(x+1)^{2}(x+2)}
= \frac{4x(x+1) + 6(x+1)}{(x+1)^{2}(x+2)}
= \frac{(4x+6)(x+1)}{(x+1)^{2}(x+2)}
Cancelling (x+1) from both numerator and denominator. Then the equation will be written as follows.
\frac{(4x+6)(x+1)}{(x+1)^{2}(x+2)}
= \frac{(4x+6)}{(x+1)(x+2)}
The numerator of simplified sum is (4x+6). Therefore answer choice C)4x+6 is the answer.