Question
Answer
$$(y-x)(z-x)(z-y)=-x^2y+x^2z+xy^2-xz^2-y^2z+yz^2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(y-x)(z-x)(z-y)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1yz-xy-xz+x^2)(z-y) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-x^2y+x^2z+xy^2-xz^2-y^2z+yz^2\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{y-x}\right) $ by each term in $ \left( z-x\right) $. $$ \left( \color{blue}{y-x}\right) \cdot \left( z-x\right) = yz-xy-xz+x^2 $$ |
| ② | Multiply each term of $ \left( \color{blue}{yz-xy-xz+x^2}\right) $ by each term in $ \left( z-y\right) $. $$ \left( \color{blue}{yz-xy-xz+x^2}\right) \cdot \left( z-y\right) = \\ = yz^2-y^2z -\cancel{xyz}+xy^2-xz^2+ \cancel{xyz}+x^2z-x^2y $$ |
| ③ | Combine like terms: $$ yz^2-y^2z \, \color{blue}{ -\cancel{xyz}} \,+xy^2-xz^2+ \, \color{blue}{ \cancel{xyz}} \,+x^2z-x^2y = -x^2y+x^2z+xy^2-xz^2-y^2z+yz^2 $$ |
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