Improper algebraic fractions

Objective. Divide first, then decompose the remainder.

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Learn: the key idea

Improper algebraic fractions have numerator degree greater than or equal to the denominator, so must first be divided to extract a polynomial part, leaving a proper remainder fraction to decompose normally. Example: (x²+3x+1)/(x+1) divides to give x+2 remainder −1, so equals x+2−1/(x+1). Only after this division can repeated or distinct linear factor methods be applied to the remaining proper fraction.

Worked example

Express the improper algebraic fraction (2x²+3x−1)/(x−1) as a polynomial plus a proper fraction.

1 $Divide 2x²+3x-1 by (x-1) using long division$
2 $2x²\divx=2x, multiply (x-1) by 2x giving 2x²-2x, subtract to leave 5x-1$
3 $5x\divx=5, multiply (x-1) by 5 giving 5x-5, subtract to leave remainder 4$
4 $Write the result as quotient plus remainder over divisor$

Answer: 2x+5+4/(x−1)

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Common mistake. Students try to apply partial fraction rules directly to an improper fraction without dividing first, producing a decomposition that cannot recombine to the original expression.

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