Question
If $x + y + z = 10$, $x^2 + y^2 + z^2 = 38$, $5xyz = 150$ and $(xy + yz + zx) = 31$, then what is the value of $2(x^3 + y^3 + z^3)$?
The correct answer is
320
We are asked to find the value of $2(x^3 + y^3 + z^3)$ given the following information:
- $x + y + z = 10$
- $x^2 + y^2 + z^2 = 38$
- $5xyz = 150$, which simplifies to $xyz = 30$
- $xy + yz + zx = 31$
Algebraic Identity for Sum of Cubes
We use the algebraic identity:
$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - (xy + yz + zx))$
Substituting Given Values
Substitute the known values into the identity:
$x^3 + y^3 + z^3 - 3(30) = (10)(38 - 31)$
Calculating $x^3 + y^3 + z^3$
Simplify the equation:
- $x^3 + y^3 + z^3 - 90 = (10)(7)$
- $x^3 + y^3 + z^3 - 90 = 70$
- $x^3 + y^3 + z^3 = 70 + 90$
- $x^3 + y^3 + z^3 = 160$
Final Calculation
The question asks for the value of $2(x^3 + y^3 + z^3)$.
- $2(x^3 + y^3 + z^3) = 2(160)$
- $2(x^3 + y^3 + z^3) = 320$
Thus, the value of $2(x^3 + y^3 + z^3)$ is 320.
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