What is the numerator of the simplified sum?

x/x^2 + 3x+2 + 3/x+1

Question:

What is the numerator of the simplified sum?

x/x^2 + 3x+2 + 3/x+1

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).