Partial Fractions: (5x − 4)/(2x² + x − 1)
(5x − 4)/(2x² + x − 1) decomposes into 3/(x + 1) + (-1/2)/(x - 1/2). A worked non-monic denominator example with the factorisation, the coefficient system, and an editable calculator.
(5x − 4)/(2x² + x − 1) = 3/(x + 1) + (-1/2)/(x - 1/2). A leading coefficient other than 1 produces fractional roots, but the method is unchanged.
e.g. 3x+11
Expanded or factored, e.g. (x-2)(x+3) or x^2+x-6
The decomposition
(5x − 4)/(2x² + x − 1) = 3/(x + 1) + (-1/2)/(x - 1/2)
Denominator factors: (x + 1) × (x - 1/2)
This is a non-monic denominator problem. A leading coefficient other than 1 produces fractional roots, but the method is unchanged.
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Frequently asked questions
How does (5x − 4)/(2x² + x − 1) decompose? ▾
(5x − 4)/(2x² + x − 1) = 3/(x + 1) + (-1/2)/(x - 1/2). A leading coefficient other than 1 produces fractional roots, but the method is unchanged.
How was the denominator factored? ▾
It factors as (x + 1) × (x - 1/2), 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.
About this Calculator
(5x − 4)/(2x² + x − 1) decomposes into 3/(x + 1) + (-1/2)/(x - 1/2). A worked non-monic denominator example with the factorisation, the coefficient system, and an editable calculator.