Multiple choice

If α and β are the roots of the equation px2 + qx + r = 0, then what is the value of α3 + β3?

  1. ( 1 p 3 ) (3pqr - q3)

  2. (3pqr - q3)

  3. p3(3pqr - q3)

  4. 3pqr
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using α + β = -q/p and αβ = r/p, α^3 + β^3 = (α + β)^3 - 3αβ(α + β). This gives (3pqr - q^3)/p^3, matching option A.

AI explanation

For the equation px squared plus qx plus r equals 0, the sum of the roots (alpha plus beta) is negative q divided by p, and the product (alpha beta) is r divided by p. Using the algebraic identity for cubes, alpha cubed plus beta cubed equals (alpha plus beta) multiplied by (alpha squared minus alpha beta plus beta squared), which simplifies to (alpha plus beta) times ((alpha plus beta) squared minus 3 alpha beta). Substituting the sum and product gives (negative q divided by p) times ((q squared divided by p squared) minus (3r divided by p)), resulting in (1 divided by p cubed) times (3pqr minus q cubed).

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