Find the value of $x^{3} + y^{3} + z^{3} - 3 x y z$ if $x + y + z = 12$ and $x^{2} + y^{2} + z^{2} = 70$ .

Open in App

Solution

We have,
$x + y + z = 12$ ............ (1)
$x^{2} + y^{2} + z^{2} = 70$ .......... (2)

On squaring equation no. (1), we get
$x^{2} + y^{2} + z^{2} + 2 (x y + y z + x z) = 144$ ......... (3)

From equation (2) and (3), we get
$70 + 2 (x y + y z + x z) = 144$
$x y + y z + x z = 37$ .......... (4)

We know that the formula,
$x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - x z)$

Put the all value in above formula, we get
$x^{3} + y^{3} + z^{3} - 3 x y z = 12 \times (70 - 37)$
$= 12 \times 33$
$= 396$

Hence, this is the correct answer.

Suggest Corrections

Similar questions

x^{3} + y^{3} + z^{3} - 3 x y z = \frac{1}{2} (x + y + z) [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]

x^{3} + y^{3} + z^{3} - 3 x y z = \frac{1}{2} (x + y + z) [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]

Verify that : $x^{3} + y^{3} + z^{3} - 3 x y z = \frac{1}{2} (x + y + z) [{(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2})]$

x^{3} + y^{3} + z^{3} - 3 x y z = \frac{1}{2} (x + y + z) [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]

x^{3} + y^{3} + z^{3} - 3 x y z = \frac{1}{2} (x + y + z) [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]

Join BYJU'S Learning Program