Partial Fractions: (x² + 2x)/((x + 1)(x − 2))
(x² + 2x)/((x + 1)(x − 2)) decomposes into 1 + (1/3)/(x + 1) + (8/3)/(x - 2). A worked improper fraction example with the factorisation, the coefficient system, and an editable calculator.
(x² + 2x)/((x + 1)(x − 2)) = 1 + (1/3)/(x + 1) + (8/3)/(x - 2). Equal degrees top and bottom means a constant falls out of the division.
e.g. 3x+11
Expanded or factored, e.g. (x-2)(x+3) or x^2+x-6
The decomposition
(x² + 2x)/((x + 1)(x − 2)) = 1 + (1/3)/(x + 1) + (8/3)/(x - 2)
Denominator factors: (x + 1) × (x - 2)
This is a improper fraction problem. Equal degrees top and bottom means a constant falls out of the division.
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Frequently asked questions
How does (x² + 2x)/((x + 1)(x − 2)) decompose? ▾
(x² + 2x)/((x + 1)(x − 2)) = 1 + (1/3)/(x + 1) + (8/3)/(x - 2). Equal degrees top and bottom means a constant falls out of the division.
How was the denominator factored? ▾
It factors as (x + 1) × (x - 2), which sets the shape of the decomposition.
How do I check the answer? ▾
Add the resulting fractions back over a common denominator — you should recover the original expression. Substituting a convenient value of x into both sides is a quicker spot check.
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(x² + 2x)/((x + 1)(x − 2)) decomposes into 1 + (1/3)/(x + 1) + (8/3)/(x - 2). A worked improper fraction example with the factorisation, the coefficient system, and an editable calculator.