A-Level Mathematics May/June 2024 Q5: Express (6x²-2x+2)/((x-1)(2x+1)) in partial fractions.

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  The given expression is an improper fraction because the degree of the numerator (2) is equal to the degree of the denominator (2). Therefore, we must include a constant term in the partial fraction decomposition.

  Let the expression be written in the form: $\frac{6x^2 - 2x + 2}{(x-1)(2x+1)} \equiv A + \frac{B}{x-1} + \frac{C}{2x+1}$

  To find the constants A, B, and C, we create an identity by multiplying by the common denominator, (x-1)(2x+1): $6x^2 - 2x + 2 \equiv A(x-1)(2x+1) + B(2x+1) + C(x-1)$

  We can find the constants by substituting convenient values for x and by comparing coefficients.

  First, let's find the constant A by comparing the coefficients of the $x^2$ term. On the left-hand side (LHS), the coefficient is 6. On the right-hand side (RHS), the $x^2$ term only comes from $A(x-1)(2x+1) = A(2x^2 - x - 1)$. $6 = 2A \implies A = 3$

  Next, to find B, let $x = 1$: $6(1)^2 - 2(1) + 2 = A(0) + B(2(1)+1) + C(0)$ $6 - 2 + 2 = B(3)$ $6 = 3B \implies B = 2$

  To find C, let $x = -\frac{1}{2}$: $6(-\frac{1}{2})^2 - 2(-\frac{1}{2}) + 2 = A(0) + B(0) + C(-\frac{1}{2}-1)$ $6(\frac{1}{4}) + 1 + 2 = C(-\frac{3}{2})$ $\frac{3}{2} + 3 = -\frac{3}{2}C$ $\frac{9}{2} = -\frac{3}{2}C$ $9 = -3C \implies C = -3$

  Therefore, the expression in partial fractions is: $3 + \frac{2}{x-1} - \frac{3}{2x+1}$

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  How the marks are awarded

  • B1 — Stating or implying the correct form for an improper fraction, $A + \frac{B}{x-1} + \frac{C}{2x+1}$.
  • M1 — Using a correct method to find at least one constant. This is shown by substituting $x=1$ or $x=-1/2$, or by expanding and comparing coefficients.
  • A1 — Obtaining one of the constants correctly, for example, $A=3$ by comparing coefficients of $x^2$.
  • A1 — Obtaining a second constant correctly, for example, $B=2$ by substituting $x=1$.
  • A1 — Obtaining the third and final constant correctly, $C=-3$, and thus determining the complete partial fraction expression.

  Common mistakes

  • Failing to recognise the expression as an improper fraction and omitting the constant term 'A', attempting to use the form $\frac{B}{x-1} + \frac{C}{2x+1}$ instead.
  • Making an arithmetic error when substituting $x = -1/2$, particularly when squaring the negative fraction or adding the resulting terms.
  • Incorrectly determining the value of A by comparing coefficients, for example by failing to expand A(x-1)(2x+1) correctly to $A(2x^2-x-1)$.
  • A sign error when solving for a constant, most commonly finding $C=3$ instead of $C=-3$.

  Examiner tip: Always check the degrees of the numerator and denominator first to determine if the algebraic fraction is proper or improper before attempting to find partial fractions.

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