All subjects · Further Mathematics · WFM01 Further Pure Mathematics 1

Expressions in the roots

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

Many expressions in α and β are symmetric: swapping α and β leaves them unchanged. Every such expression can be written using only α + β and αβ, so you can evaluate it without solving the equation.

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Key identities

Method

1. Write down α + β and αβ from the equation.

2. Rewrite the expression using an identity. Combine fractions over a common denominator first.

3. Substitute and simplify. Keep exact fractions.

Why bother?

Exam questions say "without solving the equation". Finding α and β with the formula and substituting gets no marks for the method, and surds make it slow and error-prone.

Worked example

α and β are the roots of 2x2 - 3x - 4 = 0. Find α3 + β3.

  1. α + β = 3/2 and αβ = -2.
  2. α3 + β3 = (α + β)3 - 3αβ(α + β) = 27/8 - 3(-2)(3/2).
  3. = 27/8 + 9 = 99/8

Answer: α3 + β3 = 99/8

Key idea: Rewrite every symmetric expression in terms of α + β and αβ, then substitute.

Check yourself

  1. α and β are the roots of 2x2 - 3x - 4 = 0. Find α2 + β2. Give your answer as a fraction in its simplest form.
    Show answer

    25/4. (α + β)2 - 2αβ = 9/4 + 4 = 25/4.

  2. α and β are the roots of 2x2 - 3x - 4 = 0. Find 1/α + 1/β. Give your answer as a fraction in its simplest form.
    Show answer

    -3/4. (α + β)/(αβ) = (3/2)/(-2) = -3/4.

  3. α and β are the roots of 2x2 - 3x - 4 = 0. Find (α - β)2. Give your answer as a fraction in its simplest form.
    Show answer

    41/4. (α + β)2 - 4αβ = 9/4 + 8 = 41/4.

  4. α and β are the roots of 2x2 - 3x - 4 = 0. Find α2 β + αβ2.
    Show answer

    -3. αβ(α + β) = (-2)(3/2) = -3.

  5. α and β are the roots of 2x2 - 3x - 4 = 0. Find α/β + β/α. Give your answer as a fraction in its simplest form.
    Show answer

    -25/8. (α2 + β2)/(αβ) = (25/4)/(-2) = -25/8.

  6. Which identity is correct?
    • α3 + β3 = (α + β)3 - 3αβ
    • α3 + β3 = (α + β)3
    • α3 + β3 = (α + β)3 - 3αβ(α + β)
    • α3 + β3 = (α + β)3 + 3αβ(α + β)
    Show answer

    α3 + β3 = (α + β)3 - 3αβ(α + β). (α + β)3 = α3 + 3α2β + 3αβ2 + β3 = α3 + β3 + 3αβ(α + β). Rearrange.

  7. α and β are the roots of x2 + 4x + 1 = 0. Find α3 + β3.
    Show answer

    -52. α + β = -4 and αβ = 1. (α + β)3 - 3αβ(α + β) = -64 + 12 = -52.

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Last updated 8 November 2026. Spotted a mistake? Tell us.