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Question-191798

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Question Number 191798 by Abdullahrussell last updated on 30/Apr/23
Commented by Frix last updated on 30/Apr/23
Qu 191753; just add −3+24=21
$$\mathrm{Qu}\:\mathrm{191753};\:\mathrm{just}\:\mathrm{add}\:−\mathrm{3}+\mathrm{24}=\mathrm{21} \\ $$
Commented by BaliramKumar last updated on 01/May/23
−3×24 = −72 👇
$$−\mathrm{3}×\mathrm{24}\:=\:−\mathrm{72}\:\:\: \\ $$👇
Answered by mr W last updated on 30/Apr/23
x^2 y+y^2 z+z^2 x+xy^2 +yz^2 +zx^2 =xy(x+y)+yz(y+z)+zx(z+x) =xy(x+y+z)+yz(y+z+x)+zx(z+x+y)−3xyz =(x+y+z)(xy+yz+zx)−3xyz =6×3−3×(−1) =21 ✓
$${x}^{\mathrm{2}} {y}+{y}^{\mathrm{2}} {z}+{z}^{\mathrm{2}} {x}+{xy}^{\mathrm{2}} +{yz}^{\mathrm{2}} +{zx}^{\mathrm{2}} \\ $$$$={xy}\left({x}+{y}\right)+{yz}\left({y}+{z}\right)+{zx}\left({z}+{x}\right) \\ $$$$={xy}\left({x}+{y}+{z}\right)+{yz}\left({y}+{z}+{x}\right)+{zx}\left({z}+{x}+{y}\right)−\mathrm{3}{xyz} \\ $$$$=\left({x}+{y}+{z}\right)\left({xy}+{yz}+{zx}\right)−\mathrm{3}{xyz} \\ $$$$=\mathrm{6}×\mathrm{3}−\mathrm{3}×\left(−\mathrm{1}\right) \\ $$$$=\mathrm{21}\:\checkmark \\ $$
Answered by BaliramKumar last updated on 01/May/23
(x^2 y+y^2 z+z^2 x)(xy^2 +yz^2 +zx^2 ) = {x^4 yz+xy^4 z+xyz^4 }+{(xy)^3 +(yz)^3 +(zx)^3 }+{3(xyz)^2 } = {−x^3 −y^3 −z^3 }+{(xy+yz+zx)((xy+yz+zx)^2 −3(xy^2 z+xyz^2 +x^2 yz))+3x^2 y^2 z^2 }+{3} = −{(x+y+z)((x+y+z)^2 −3(xy+yz+zx))+3xyz}+{3(3^2 −3(−x−y−z))+3(−1)^2 } +{ 3} = −{6(6^2 −3×3)+3(−1)}+{3(9−3(−6))+3}+{3} = −{159}+{84}+{3} = determinant (((−72))) p+q = 21 pq = −72 s^2 −(p+q)s+(pq) = 0 s^2 −21s−72=0 s = −3 or 24
$$\left({x}^{\mathrm{2}} {y}+{y}^{\mathrm{2}} {z}+{z}^{\mathrm{2}} {x}\right)\left({xy}^{\mathrm{2}} +{yz}^{\mathrm{2}} +{zx}^{\mathrm{2}} \right) \\ $$$$=\:\left\{{x}^{\mathrm{4}} {yz}+{xy}^{\mathrm{4}} {z}+{xyz}^{\mathrm{4}} \right\}+\left\{\left({xy}\right)^{\mathrm{3}} +\left({yz}\right)^{\mathrm{3}} +\left({zx}\right)^{\mathrm{3}} \right\}+\left\{\mathrm{3}\left({xyz}\right)^{\mathrm{2}} \right\}\:\:\:\:\:\:\:\:\: \\ $$$$=\:\left\{−{x}^{\mathrm{3}} −{y}^{\mathrm{3}} −{z}^{\mathrm{3}} \right\}+\left\{\left({xy}+{yz}+{zx}\right)\left(\left({xy}+{yz}+{zx}\right)^{\mathrm{2}} −\mathrm{3}\left({xy}^{\mathrm{2}} {z}+{xyz}^{\mathrm{2}} +{x}^{\mathrm{2}} {yz}\right)\right)+\mathrm{3}{x}^{\mathrm{2}} {y}^{\mathrm{2}} {z}^{\mathrm{2}} \right\}+\left\{\mathrm{3}\right\}\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$=\:−\left\{\left({x}+{y}+{z}\right)\left(\left({x}+{y}+{z}\right)^{\mathrm{2}} −\mathrm{3}\left({xy}+{yz}+{zx}\right)\right)+\mathrm{3}{xyz}\right\}+\left\{\mathrm{3}\left(\mathrm{3}^{\mathrm{2}} −\mathrm{3}\left(−{x}−{y}−{z}\right)\right)+\mathrm{3}\left(−\mathrm{1}\right)^{\mathrm{2}} \right\}\:+\left\{\:\mathrm{3}\right\}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$=\:−\left\{\mathrm{6}\left(\mathrm{6}^{\mathrm{2}} −\mathrm{3}×\mathrm{3}\right)+\mathrm{3}\left(−\mathrm{1}\right)\right\}+\left\{\mathrm{3}\left(\mathrm{9}−\mathrm{3}\left(−\mathrm{6}\right)\right)+\mathrm{3}\right\}+\left\{\mathrm{3}\right\}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$=\:−\left\{\mathrm{159}\right\}+\left\{\mathrm{84}\right\}+\left\{\mathrm{3}\right\} \\ $$$$=\:\begin{array}{|c|}{−\mathrm{72}}\\\hline\end{array} \\ $$$$ \\ $$$${p}+{q}\:=\:\mathrm{21}\:\:\:\:\:\:\:\:{pq}\:=\:−\mathrm{72} \\ $$$${s}^{\mathrm{2}} −\left({p}+{q}\right){s}+\left({pq}\right)\:=\:\mathrm{0} \\ $$$${s}^{\mathrm{2}} −\mathrm{21}{s}−\mathrm{72}=\mathrm{0} \\ $$$${s}\:=\:−\mathrm{3}\:{or}\:\mathrm{24} \\ $$$$ \\ $$

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