α^3 + β^3 equals which identity?

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Boost your skills in A Level Further Mathematics Core Pure. Study with flashcards and multiple choice questions, each question is followed by hints and explanations. Get prepared for your exam with confidence!

Multiple Choice

α^3 + β^3 equals which identity?

Explanation:
When you expand (α+β)^3, you get α^3 + 3α^2β + 3αβ^2 + β^3. The middle terms combine to 3αβ(α+β), so (α+β)^3 = α^3 + β^3 + 3αβ(α+β). Rearranging gives α^3 + β^3 = (α+β)^3 − 3αβ(α+β). This is the form that matches α^3 + β^3 in terms of (α+β) and αβ. The other options alter the cross terms or drop the (α+β) factor, which does not hold in general (a quick numeric check shows they'll fail).

When you expand (α+β)^3, you get α^3 + 3α^2β + 3αβ^2 + β^3. The middle terms combine to 3αβ(α+β), so (α+β)^3 = α^3 + β^3 + 3αβ(α+β). Rearranging gives α^3 + β^3 = (α+β)^3 − 3αβ(α+β). This is the form that matches α^3 + β^3 in terms of (α+β) and αβ. The other options alter the cross terms or drop the (α+β) factor, which does not hold in general (a quick numeric check shows they'll fail).

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