Question : $x, y,$ and $z$ are real numbers. If $x^{3} + y^{3} + z^{3} = 13, x + y + z = 1$ and $x y z = 1$ , then what is the value of $x y + y z + z x$ ?
Option 1: –1
Option 2: 1
Option 3: 3
Option 4: –3

Question

Question : If $(x + y + z) = 12$ , $x y + y z + z x = 44$ , and $x y z = 48$ , then what is the value of $x^{3} + y^{3} + z^{3}$ ?
Option 1: 104
Option 2: 144
Option 3: 196
Option 4: 288

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Answer 1

Correct Answer: –3

Solution : Given: $x^{3} + y^{3} + z^{3} = 13$ , $x + y + z = 1$ , $x y z = 1$
We know, $x^{3} + y^{3} + z^{3} - 3 x y z = (x^{2} + y^{2} + z^{2} - x y - y z - z x)$
$\Rightarrow x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) ((x + y + z)^{2} - 3 (x y + y z + z x))$
Substituting values of $x^{3} + y^{3} + z^{3}$ , $x + y + z$ and $x y z$ ,
$\Rightarrow 13 - 3 (1) = (1) ((1^{2} - 3 (x y + y z + z x))$
$\Rightarrow x y + y z + z x = \frac{10 - 1}{- 3}$
$∴ x y + y z + z x = - 3$
Hence, the correct answer is –3.

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Question : x,y, and z are real numbers. If x3+y3+z3=13,x+y+z=1 and xyz=1, then what is the value of xy+yz+zx?Option 1: –1Option 2: 1Option 3: 3Option 4: –3

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Question : $x, y,$ and $z$ are real numbers. If $x^{3} + y^{3} + z^{3} = 13, x + y + z = 1$ and $x y z = 1$ , then what is the value of $x y + y z + z x$ ?
Option 1: –1
Option 2: 1
Option 3: 3
Option 4: –3