Mister Exam
Least common denominator (n*n-2*n)/(n+1)-(n*n-1)/(n+2)
The solution
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n*n - 2*n n*n - 1 --------- - ------- n + 1 n + 2
$$\frac{- 2 n + n n}{n + 1} - \frac{n n - 1}{n + 2}$$
(n*n - 2*n)/(n + 1) - (n*n - 1)/(n + 2)
General simplification
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2
1 - n - 3*n
------------
2
2 + n + 3*n
$$\frac{- n^{2} - 3 n + 1}{n^{2} + 3 n + 2}$$
(1 - n^2 - 3*n)/(2 + n^2 + 3*n)
Fraction decomposition
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-1 - 3/(2 + n) + 3/(1 + n)
$$-1 - \frac{3}{n + 2} + \frac{3}{n + 1}$$
3 3
-1 - ----- + -----
2 + n 1 + n
Assemble expression
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2 2 n - 2*n -1 + n -------- - ------- 1 + n 2 + n
$$- \frac{n^{2} - 1}{n + 2} + \frac{n^{2} - 2 n}{n + 1}$$
(n^2 - 2*n)/(1 + n) - (-1 + n^2)/(2 + n)
Rational denominator
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/ 2\ / 2 \
(1 + n)*\1 - n / + (2 + n)*\n - 2*n/
-------------------------------------
(1 + n)*(2 + n)
$$\frac{\left(1 - n^{2}\right) \left(n + 1\right) + \left(n + 2\right) \left(n^{2} - 2 n\right)}{\left(n + 1\right) \left(n + 2\right)}$$
((1 + n)*(1 - n^2) + (2 + n)*(n^2 - 2*n))/((1 + n)*(2 + n))
Numerical answer
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(n^2 - 2.0*n)/(1.0 + n) - (-1.0 + n^2)/(2.0 + n)
(n^2 - 2.0*n)/(1.0 + n) - (-1.0 + n^2)/(2.0 + n)
Powers
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2 2 n - 2*n -1 + n -------- - ------- 1 + n 2 + n
$$- \frac{n^{2} - 1}{n + 2} + \frac{n^{2} - 2 n}{n + 1}$$
2 2 n - 2*n 1 - n -------- + ------ 1 + n 2 + n
$$\frac{1 - n^{2}}{n + 2} + \frac{n^{2} - 2 n}{n + 1}$$
(n^2 - 2*n)/(1 + n) + (1 - n^2)/(2 + n)
Common denominator
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3
-1 + ------------
2
2 + n + 3*n
$$-1 + \frac{3}{n^{2} + 3 n + 2}$$
-1 + 3/(2 + n^2 + 3*n)
Trigonometric part
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2 2 n - 2*n -1 + n -------- - ------- 1 + n 2 + n
$$- \frac{n^{2} - 1}{n + 2} + \frac{n^{2} - 2 n}{n + 1}$$
(n^2 - 2*n)/(1 + n) - (-1 + n^2)/(2 + n)
Combining rational expressions
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/ 2\
- (1 + n)*\-1 + n / + n*(-2 + n)*(2 + n)
----------------------------------------
(1 + n)*(2 + n)
$$\frac{n \left(n - 2\right) \left(n + 2\right) - \left(n + 1\right) \left(n^{2} - 1\right)}{\left(n + 1\right) \left(n + 2\right)}$$
(-(1 + n)*(-1 + n^2) + n*(-2 + n)*(2 + n))/((1 + n)*(2 + n))
Combinatorics
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/ 2 \ -\-1 + n + 3*n/ ----------------- (1 + n)*(2 + n)
$$- \frac{n^{2} + 3 n - 1}{\left(n + 1\right) \left(n + 2\right)}$$
-(-1 + n^2 + 3*n)/((1 + n)*(2 + n))