A) Only \[(y-z)\]
B) Only \[(z-x)\]
C) Both \[(y-z)\] and \[(z-x)\]
D) Neither \[(y-z)\] nor \[(z-x)\]
Correct Answer: C
Solution :
\[x({{y}^{2}}-{{z}^{2}})+y({{z}^{2}}-{{x}^{2}})+z({{x}^{2}}-{{y}^{2}})\] If it is divisible by (y - z), then \[yz=0\Rightarrow ~y=z\] On putting y = z, we get \[x({{z}^{2}}-{{z}^{2}})+z({{z}^{2}}-{{x}^{2}})+z({{x}^{2}}-{{z}^{2}})\] = \[{{z}^{3}}-z{{x}^{2}}+z{{x}^{2}}-{{z}^{3}}=0\] Hence, y - z is a factor, so it is divisible by (y-z). Also, if z - x is a factor, then \[\Rightarrow z-x=0\Rightarrow z=x\] On putting z = x, we get \[\Rightarrow x({{y}^{2}}-{{x}^{2}})+y({{y}^{2}}-{{x}^{2}})+x({{x}^{2}}-{{y}^{2}})\] \[x{{y}^{2}}-{{x}^{3}}+{{x}^{3}}-x{{y}^{2}}=\] [0]
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