If α and β are the roots of $x^2-8x+5=0$, then α³ + β³ equals:

Here α + β = 8 and αβ = 5. Using α³ + β³ = $(\alpha +\beta {)}^3-3$αβ(α + β) = $512-3(5)(8)=512-120=392$. (512 forgets the correction term altogether, 497 subtracts only 3αβ = 15 without the factor (α + β), and 432 uses the coefficient 2 that belongs to α² + β².) Directly, the roots are 4 ± √11, whose cubes are 196 + 59√11 and 196 − 59√11; the irrational parts cancel and the sum is 392.