Let-x-y-z-be-real-numbers-satisfying-the-equations-x-y-z-7-xy-yz-zx-10-xyz-6-Find-the-value-of-x-3-y-3-z-3-

Question Number 217377 by ArshadS last updated on 12/Mar/25

Let x, y, z be real numbers satisfying the equations

x + y + z = 7

xy + yz + zx = 10

xyz = 6

Find the value of

x^{3} + y^{3} + z^{3}

Answered by Rasheed.Sindhi last updated on 12/Mar/25

{\begin{cases} x + y + z = 7 \dots (i) \\ xy + yz + zx = 10 \dots (i i) \\ xyz = 6 \dots (i i i) \\ x^{3} + y^{3} + z^{3} = ? \end{cases}

{(i)}^{2} : x^{2} + y^{2} + z^{2} + 2 (xy + yz + zx) = 49

x^{2} + y^{2} + z^{2} + 2 (10) = 49

x^{2} + y^{2} + z^{2} = 29

x^{3} + y^{3} + z^{3} - 3 xyz = (x + y + z) (x^{2} + y^{2} + z^{2} - (xy + yz + zx))

x^{3} + y^{3} + z^{3} - 3 (6) = (7) (29 - (10))

x^{3} + y^{3} + z^{3} = 18 + (7) (29 - (10))

= 18 + 133 = 151

Commented by ArshadS last updated on 13/Mar/25

T h a n k s s i r!

Answered by mr W last updated on 12/Mar/25

{(x + y + z)}^{3} = x^{3} + y^{3} + z^{3} - 3 x y z + 3 (x + y + z) (x y + y z + z x)

7^{3} = x^{3} + y^{3} + z^{3} - 3 \times 6 + 3 \times 7 \times 10

\Rightarrow x^{3} + y^{3} + z^{3} = 7^{3} + 3 \times 6 - 3 \times 7 \times 10 = 151

Commented by ArshadS last updated on 13/Mar/25

Thanks sir!

Facebook Tweet Pin