Solve n/n+2-n-1/n+1-2/left(n+1right)left(n+2right)

$$\frac{ n }{ n+2 } - \frac{ n-1 }{ n+1 } - \frac{ 2 }{ \left( n+1 \right) \left( n+2 \right) }$$

Evaluate

$$0$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of n+2 and n+1 is \left(n+1\right)\left(n+2\right). Multiply \frac{n}{n+2} times \frac{n+1}{n+1}. Multiply \frac{n-1}{n+1} times \frac{n+2}{n+2}.

$$\frac{n\left(n+1\right)}{\left(n+1\right)\left(n+2\right)}-\frac{\left(n-1\right)\left(n+2\right)}{\left(n+1\right)\left(n+2\right)}-\frac{2}{\left(n+1\right)\left(n+2\right)}$$

Since \frac{n\left(n+1\right)}{\left(n+1\right)\left(n+2\right)} and \frac{\left(n-1\right)\left(n+2\right)}{\left(n+1\right)\left(n+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{n\left(n+1\right)-\left(n-1\right)\left(n+2\right)}{\left(n+1\right)\left(n+2\right)}-\frac{2}{\left(n+1\right)\left(n+2\right)}$$

Do the multiplications in n\left(n+1\right)-\left(n-1\right)\left(n+2\right).

$$\frac{n^{2}+n-n^{2}-2n+n+2}{\left(n+1\right)\left(n+2\right)}-\frac{2}{\left(n+1\right)\left(n+2\right)}$$

Combine like terms in n^{2}+n-n^{2}-2n+n+2.

$$\frac{2}{\left(n+1\right)\left(n+2\right)}-\frac{2}{\left(n+1\right)\left(n+2\right)}$$

Since \frac{2}{\left(n+1\right)\left(n+2\right)} and \frac{2}{\left(n+1\right)\left(n+2\right)} have the same denominator, subtract them by subtracting their numerators. Subtract 2 from 2 to get 0.

$$\frac{0}{\left(n+1\right)\left(n+2\right)}$$

Zero divided by any non-zero term gives zero.

$$0$$

Factor

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