Question

Use the identity x^3+y^3+z^3−3xyz=(x+y+z)(x^2+y^2+z^2−x^y−y^z−z^x) to determine the value of the sum of three integers given:

the sum of their squares is 110,
the sum of their cubes is 684,
the product of the three integers is 210,
and the sum of any two products (xy+yz+zx) is 107.

Answer 1

Correctly written, your identity would tell you ...

... (sum of cubes) - 3·(product) = (sum of integers)·((sum of squares) - (sum of any two products))

Filling in the given numbers, you have ...

... 684 - 3·210 = (sum of integers)·(110 -107)

... 54 = (sum of integers)·3 . . . . . simplify

... sum of integers = 54/3 . . . . . . divide by the coefficient of the variable

... sum of integers = 18

_____

Integers 5, 6, and 7 meet these conditions.