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Partial fractions questions

This lesson runs the addition of algebraic fractions backwards, splitting one awkward fraction into simpler pieces with unknown constants on top. The constants are found by substituting well-chosen values of x or by comparing coefficients, and the pieces are what integration and series expansions need later.

6 original questions · 24 marks · the partial fractions notes · Algebra and functions

Every question here is written for this library rather than taken from a past paper. Write your answer out before opening the worked one: the answers award marks point by point, and the marks are easier to see when you have something of your own to compare against.

• 3 marks · warm-up

  Express (3x + 5)/((x + 1)(x + 2)) in partial fractions.

  Worked answer

  Set the fraction equal to A/(x + 1) + B/(x + 2) and multiply up, giving 3x + 5 = A(x + 2) + B(x + 1). Substituting x = −1 kills the B bracket and leaves 2 = A. Substituting x = −2 leaves −1 = −B, so B = 1. The answer is 2/(x + 1) + 1/(x + 2). B1 for the correct form, M1 for the substitutions and A1 for the two constants, so a candidate who writes the form and then stalls still banks one.

• 2 marks · warm-up

  Writing (3x + 5)/((x + 1)(x + 2)) as A/(x + 1) + B/(x + 2) gives 3x + 5 = A(x + 2) + B(x + 1). Explain why substituting x = −1 finds A immediately.

  Worked answer

  At x = −1 the factor (x + 1) is zero, so the whole B term vanishes and the equation collapses to a statement about A alone. Each root of the denominator switches off every constant except its own. B1 B1 for the vanishing factor and for the equation in A alone. Comparing coefficients reaches the same values, but it needs simultaneous equations where substitution needs none.

• 4 marks

  Express (6x² + 5x − 2)/(x(x − 1)(x + 2)) in partial fractions.

  Worked answer

  Set A/x + B/(x − 1) + C/(x + 2) and multiply up: 6x² + 5x − 2 = A(x − 1)(x + 2) + Bx(x + 2) + Cx(x − 1). Then x = 0 gives −2 = −2A, so A = 1; x = 1 gives 9 = 3B, so B = 3; x = −2 gives 12 = 6C, so C = 2. The answer is 1/x + 3/(x − 1) + 2/(x + 2), and any spare x value, say x = 2, checks it: 32 = 4 + 24 + 4. B1 for the form, M1 for the substitutions, A1 for A = 1 and B = 3, A1 for C = 2.

• 4 marks

  Express (2x + 7)/(x + 3)² in partial fractions.

  Worked answer

  A squared bracket needs both powers, so try A/(x + 3) + B/(x + 3)². Multiplying up gives 2x + 7 = A(x + 3) + B. Substituting x = −3 leaves 1 = B, and comparing the x coefficients gives A = 2. The answer is 2/(x + 3) + 1/(x + 3)². B1 for including both powers, M1 for multiplying up, A1 for B = 1, A1 for A = 2. One fraction per bracket fails here, because a single A/(x + 3)² cannot produce the 2x on the left. The repeated factor demands the pair, and writing it down is the first mark.

• 5 marks

  Express (x² + 4x + 5)/((x + 1)(x + 2)) in partial fractions.

  Worked answer

  The fraction is improper: the top's degree matches the bottom's. Divide first: x² + 4x + 5 = (x² + 3x + 2) + (x + 3), so the fraction is 1 + (x + 3)/((x + 1)(x + 2)). Splitting the proper part: x = −1 gives A = 2, x = −2 gives B = −1. The answer is 1 + 2/(x + 1) − 1/(x + 2). M1 for spotting the improper fraction and dividing, A1 for the quotient and proper remainder, B1 for the partial-fraction form, A1 for A = 2, A1 for B = −1. Skipping the division and writing two fractions for an improper top is the standard wrong turn.

• 6 marks · stretch

  Express (3x² + 7x + 8)/((x + 1)²(x + 2)) in partial fractions.

  Worked answer

  Three constants: A/(x + 1) + B/(x + 1)² + C/(x + 2). Multiplying up: 3x² + 7x + 8 = A(x + 1)(x + 2) + B(x + 2) + C(x + 1)². Then x = −1 gives 4 = B; x = −2 gives 6 = C; comparing x² coefficients, A + C = 3, so A = −3. The answer is −3/(x + 1) + 4/(x + 1)² + 6/(x + 2). B1 for the three-term form, M1 for multiplying up, A1 for B = 4, A1 for C = 6, M1 for comparing x² coefficients, A1 for A = −3. A negative constant is not a mistake; the check at x = 0 confirms it: 8 = −6 + 8 + 6.

The same practice on paper: the printable workbook for this topic, questions and a worked answer book.

Practise partial fractions one question at a time

The player marks nothing for you. It shows one question, waits, then shows the worked answer so you can mark yourself, and brings a question back sooner when it went badly.