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  "body": "\n## Summary\n\nPartial fraction decomposition writes a rational function as a sum of a polynomial and simple fractions.\nLet $K$ be a field, and let $P/Q$ be a rational function with $P, Q \\in K[X]$ and $Q \\neq 0$.\nAssume that $Q = \\prod_{i=1}^{r} q_i^{m_i}$, where the $q_i$ are distinct monic irreducible polynomials.\nThen there are unique polynomials $B$ and $A_{ij}$, with $\\deg A_{ij} \u003c \\deg q_i$, such that\n$$\\frac{P}{Q} = B + \\sum_{i=1}^{r} \\sum_{j=1}^{m_i} \\frac{A_{ij}}{q_i^{\\,j}}.$$\n\n## Remarks\n\nExample: $\\frac{1}{1 - x - x^2} = \\frac{1}{\\sqrt{5}} \\left( \\frac{\\varphi}{1 - \\varphi x} - \\frac{\\psi}{1 - \\psi x} \\right)$, with $\\varphi$ and $\\psi$ the roots of $X^2 - X - 1$.\nIn combinatorics it gives the coefficients of a rational [[generating function|0HH6JX5WZZ]]. Each term $\\frac{1}{1 - ax}$ expands as a [[geometric series|0R8JR0ARJ4]].\n\n## Explanations\n\n### L1 human\n\nA complicated fraction of polynomials splits into a sum of simple fractions.\nFor example, $\\frac{1}{x(x+1)} = \\frac{1}{x} - \\frac{1}{x+1}$.\n",
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