Question : If $x + y + z =1$, $xy + yz + zx = - 26$, and $x^3+y^3+z^3 = 151$. then what will be the value of $xyz$?
Option 1: –18
Option 2: 32
Option 3: 24
Option 4: –30
Correct Answer: 24
Solution :
Given: $x + y + z =1$, $xy + yz + zx = - 26$, and $x^3+y^3+z^3 = 151$
We know,
$x^{2} + y^{2} + z^{2} = (x + y + z)^{2} − 2(xy + yz + zx)$
$\therefore x^{2} + y^{2} + z^{2} = (1)^{2} - 2(-26)= 53$
Now, $x^{3} + y^{3} + z^{3} − 3 xyz = (x + y + z )(x^{2} + y^{2} + z^{2} − xy − yz − zx)$
⇒ $151 − 3xyz = 1(x^{2} + y^{2} + z^{2} − xy − yz − zx)$
⇒ $151 − 3xyz = 1(53 − (− 26))$
⇒ $151− 3xyz = 1(79) = 79$
⇒ $3xyz = 151 - 79 = 72$
$\therefore xyz = \frac{72}{3}= 24$
Hence, the correct answer is 24.
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