An algebraic fraction is improper when the degree of the numerator is greater than or equal to the degree of the denominator. You rewrite it as a quotient plus a proper fraction, either by algebraic long division or by writing an identity and comparing coefficients. Exam questions give the target form, such as Ax + B + C/(x + 2), and ask you to find the constants.
Key ideas for improper fractions
- Improper: degree of numerator ≥ degree of denominator
- numerator ≡ (divisor)(quotient) + remainder
- Dividing by a linear factor: the remainder is a constant
- Dividing by a quadratic: the remainder has the form Cx + D
As x → ±∞ the proper-fraction part tends to 0, so the curve approaches the quotient.
What is an improper algebraic fraction?
Just like 7/3 with numbers, an algebraic fraction is improper when its numerator has degree at least as large as its denominator. It can be written as a whole part, the quotient, plus a proper fraction whose numerator has lower degree than its denominator.
This matters because many later techniques, especially partial fractions and integration, only work on proper fractions.
Algebraic long division
Divide the numerator by the denominator exactly as you would with numbers: divide the leading terms, multiply back, subtract, and bring down the next term. The result is quotient + remainder/divisor.
Keep the powers in descending order and write a zero term for any missing power, such as 0x² in 2x³ + 0x² + x − 5. Gaps are where sign slips happen.
Comparing coefficients instead of dividing
Instead of dividing, write the identity numerator ≡ (divisor)(Ax + B) + C, expand the right-hand side, and equate the coefficients of each power of x. Match the highest power first, then work down; each comparison fixes one unknown.
This is often quicker than division, and it is the natural method when the constants themselves are the answer.
Dividing by a quadratic
When the divisor is a quadratic, the quotient has degree two less than the numerator, and the remainder can be linear. So the target form looks like quotient + (Cx + D)/(quadratic). If the divisor is given as a product of brackets, multiply it out first.
Some questions run the identity backwards, giving the quotient and remainder and asking for unknown coefficients in the numerator. Multiply the quotient by the divisor, add the remainder and compare coefficients.
Oblique asymptotes from the quotient
Once the fraction is written as quotient + proper fraction, the proper fraction tends to 0 as x gets very large in either direction. So the graph approaches the quotient. A linear quotient gives an oblique (slanted) asymptote, a constant gives a horizontal one.
Worked examples
Example 1. Express (x² + 5x + 8)/(x + 2) in the form Ax + B + C/(x + 2).
- Divide: x² + 5x + 8 = (x + 2)(x + 3) + 2.
- The quotient is x + 3 and the remainder is 2.
- So A = 1, B = 3 and C = 2.
Answer: x + 3 + 2/(x + 2).
Marks: M1 for the division, A1 for A, B and C.
Example 2. Find A, B and C such that 3x² + 1 ≡ (x − 1)(Ax + B) + C.
- Expand: (x − 1)(Ax + B) + C = Ax² + (B − A)x + (C − B).
- x² terms: A = 3.
- x terms: B − A = 0, so B = 3.
- Constants: C − B = 1, so C = 4.
Answer: A = 3, B = 3, C = 4.
Marks: M1 for equating coefficients, A1 for A, B and C.
Example 3. Express (2x³ + 3x² + 2x + 1)/(x² + 1) in the form Ax + B + (Cx + D)/(x² + 1).
- Divide: (x² + 1)(2x + 3) = 2x³ + 3x² + 2x + 3.
- So 2x³ + 3x² + 2x + 1 = (x² + 1)(2x + 3) − 2.
- The quotient is 2x + 3 and the remainder is −2, so C = 0 and D = −2.
Answer: 2x + 3 − 2/(x² + 1), with A = 2, B = 3, C = 0, D = −2.
Marks: M1 for the division, A1 for all four constants.
Example 4. Write x²/(x − 1) as a quotient plus a proper fraction, and state the oblique asymptote of y = x²/(x − 1).
- Divide: x² = (x − 1)(x + 1) + 1.
- So x²/(x − 1) = x + 1 + 1/(x − 1).
- As x → ±∞, 1/(x − 1) → 0, so the curve approaches y = x + 1.
Answer: x + 1 + 1/(x − 1); asymptote y = x + 1.
Marks: M1 for dividing, A1 for the asymptote.
Common mistakes
- Leaving out missing powers in long division. Write 0x² or 0x as a placeholder so the columns line up.
- Splitting an improper fraction directly into partial fractions. Divide first; partial fractions only work on proper fractions.
- Using a constant remainder over a quadratic. When dividing by a quadratic, allow a remainder of the form Cx + D.
- Sign errors when subtracting in long division. Subtract the whole product line, and check by multiplying back.
- Comparing coefficients before expanding fully. Expand and collect the right-hand side first, then match each power of x.
How improper fractions are examined
Edexcel International A-Level tests improper fractions on the Pure 3 paper (WMA13), as part of algebraic methods. In Edexcel A-Level (9MA0) they appear in second-year Pure, usually alongside partial fractions. Questions normally:
- state an identity with constants A, B, C (and sometimes D) to be found
- ask for a quotient plus a proper fraction after dividing by a linear or quadratic expression
- give the result and ask for unknown coefficients in the numerator
- ask for the equation of an oblique asymptote
Exam-style questions with mark schemes are in the Pure 3 algebraic methods exam questions. For more practice, use the algebraic methods worksheet or the algebra revision engine.
This builds on algebraic division and feeds directly into partial fractions.
Frequently asked questions
How do I know if an algebraic fraction is improper?
Compare degrees. If the highest power in the numerator is equal to or greater than the highest power in the denominator, the fraction is improper.
Is long division or comparing coefficients better?
Both give the same answer. Comparing coefficients is often quicker when the form is given; long division is safer when you are unsure what form to expect.
What is an oblique asymptote?
A slanted straight line that a curve approaches as x gets very large. For an improper fraction it is the linear quotient.