Question 69

If x + y + z = 10 and xy + yz + zx = 15, then find the value of $$x^3 + y^3 + z^3 — 3xyz$$.

Solution

$$x^3 + y^3 + z^3 — 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - xz)$$

x + y + z = 10

Taking square on both sides,

$$(x + y + z)^2 = 100$$

$$x^2 + y^2 + z^2 + 2(xy + yz + xz) = 100$$

$$x^2 + y^2 + z^2 = 100 - 2\times 15 = 00 - 30 = 70$$

$$x^3 + y^3 + z^3 — 3xyz = (10)(70 - 15) = 10 \times 55 = 550$$

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