All subjects · Mathematics · WMA14 (P4)

Improper fractions and partial fractions

If the top has degree at least as big as the bottom, the fraction is improper. You need a polynomial part as well as the partial fractions.

Spotting improper fractions

Compare degrees after expanding the bottom. (2x2 + 2x - 3)/((x + 1)(x - 2)) has degree 2 over degree 2, so it is improper.

Method 1: divide first

Divide the top by the expanded bottom to get a quotient and a proper remainder fraction, then split the remainder into partial fractions.

Method 2: include the polynomial part

Write (2x2 + 2x - 3)/((x + 1)(x - 2)) ≡ A + B/(x + 1) + C/(x - 2). The polynomial part has degree (top degree - bottom degree): a constant here, Ax + B for a cubic over a quadratic. Compare the leading coefficients to find A.

Worked example

Express (2x2 + 2x - 3)/((x + 1)(x - 2)) in partial fractions.

  1. 2x2 + 2x - 3 ≡ A(x + 1)(x - 2) + B(x - 2) + C(x + 1)
  2. Compare x2: A = 2.
  3. x = -1: -3 = -3B, so B = 1.
  4. x = 2: 9 = 3C, so C = 3.

Answer: 2 + 1/(x + 1) + 3/(x - 2)

Key idea: Improper means degree of top ≥ degree of bottom. Divide first, or include a polynomial part of degree (top - bottom).

Check yourself

  1. Which fraction is improper?
    • (x + 1)/(x2 - 4)
    • 3/(x - 2)
    • (2x - 1)/((x + 1)(x + 3))
    • (x2 + 1)/(x2 - 4)
    Show answer

    (x2 + 1)/(x2 - 4). Top and bottom both have degree 2.

  2. (x2 + 3x + 4)/((x + 1)(x + 2)) ≡ A + B/(x + 1) + C/(x + 2). Find C.
    Show answer

    -2. x2 + 3x + 4 ≡ A(x + 1)(x + 2) + B(x + 2) + C(x + 1). x = -2: 4 - 6 + 4 = 2 = -C, so C = -2.

  3. (4x2 + 1)/((2x - 1)(x + 1)) ≡ A + B/(2x - 1) + C/(x + 1). Find A.
    Show answer

    2. The bottom expands to 2x2 + x - 1. Comparing x2 coefficients: 4 = 2A, so A = 2.

  4. What form should you use for (x3 + 2)/((x - 1)(x + 3))?
    • Ax + B + C/(x - 1) + D/(x + 3)
    • A + B/(x - 1) + C/(x + 3)
    • A/(x - 1) + B/(x + 3)
    • Ax + B/(x - 1) + C/(x + 3)
    Show answer

    Ax + B + C/(x - 1) + D/(x + 3). Cubic over quadratic: the polynomial part has degree 3 - 2 = 1, so Ax + B.

  5. (x3 + 2)/((x - 1)(x + 3)) ≡ Ax + B + C/(x - 1) + D/(x + 3). Find C. Give your answer as a fraction in its simplest form.
    Show answer

    3/4. Multiply through and put x = 1: 3 = C(1 + 3), so C = 3/4.

  6. What is a reliable first step for an improper fraction?
    • Divide the top by the bottom, or include a polynomial part
    • Split it straight into A/(x + a) + B/(x + b)
    • Cancel the x2 terms on the top and bottom
    • Differentiate the top and the bottom
    Show answer

    Divide the top by the bottom, or include a polynomial part. Without the polynomial part, the identity cannot hold for all x.

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Last updated 7 November 2026. Spotted a mistake? Tell us.