Mathematics: 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

Find the sum of the terms. x/x^2+3x+2 + 3/x+1 the

renaudciera

01.01.2022

Option (C) is the correct 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.

$\frac{(4x+6)(x+1)}{(x+1)^{2}(x+2)}$

= $\frac{(4x+6)}{(x+1)(x+2)}$

Thus, we can conclude that the numerator of simplified sum is (4x+6).

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