Find the factors of $a\left( {{b^4} - {c^4}} \right) + b\left( {{c^4} - {a^4}} \right) + c\left( {{a^4} - {b^4}} \right).$
Answer
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Hint: Try to use the formula of $\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)$.
We can write above equation as $a\left( {{b^4} - {c^4}} \right) - b\left( {{b^4} - {c^4}} \right) - b\left( {{a^4} - {b^4}} \right) + c\left( {{a^4} - {b^4}} \right).$
$ \Rightarrow \left( {{b^2} - {c^2}} \right)\left( {{b^2} + {c^2}} \right)\left( {a - b} \right) + \left( {{a^2} - {b^2}} \right)\left( {{a^2} + {b^2}} \right)\left( {c - b} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {b + c} \right)\left( {{b^2} + {c^2}} \right)\left( {a - b} \right) + \left( {a - b} \right)\left( {a + b} \right)\left( {{a^2} + {b^2}} \right)\left( {c - b} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {\left( {b + c} \right)\left( {{b^2} + {c^2}} \right) - \left( {a + b} \right)\left( {{a^2} + {b^2}} \right)} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {{b^3} + bc + c{b^2} + {c^3} - {a^3} - a{b^2} - b{a^2} - {b^3}} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {bc\left( {c + b} \right) - ab\left( {a + b} \right) + \left( {c - a} \right)\left( {{c^2} + {a^2} + ac} \right)} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {b\left( {{c^2} + cb - {a^2} - ab} \right) + \left( {c - a} \right)\left( {{c^2} + {a^2} + ac} \right)} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {b\left( {{c^2} - {a^2}} \right) + {b^2}\left( {c - a} \right)} \right) + \left( {c - a} \right)\left( {{c^2} + {a^2} + ac} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {c - a} \right)\left( {b\left( {c + a} \right) + {b^2} + {c^2} + {a^2} + ac} \right)$
Answer $ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {c - a} \right)\left( {{a^2} + {b^2} + {c^2} + ac + bc + ab} \right)$
Note: In these questions, try to solve the terms one by one by taking common out and using formulas to simplify.
We can write above equation as $a\left( {{b^4} - {c^4}} \right) - b\left( {{b^4} - {c^4}} \right) - b\left( {{a^4} - {b^4}} \right) + c\left( {{a^4} - {b^4}} \right).$
$ \Rightarrow \left( {{b^2} - {c^2}} \right)\left( {{b^2} + {c^2}} \right)\left( {a - b} \right) + \left( {{a^2} - {b^2}} \right)\left( {{a^2} + {b^2}} \right)\left( {c - b} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {b + c} \right)\left( {{b^2} + {c^2}} \right)\left( {a - b} \right) + \left( {a - b} \right)\left( {a + b} \right)\left( {{a^2} + {b^2}} \right)\left( {c - b} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {\left( {b + c} \right)\left( {{b^2} + {c^2}} \right) - \left( {a + b} \right)\left( {{a^2} + {b^2}} \right)} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {{b^3} + bc + c{b^2} + {c^3} - {a^3} - a{b^2} - b{a^2} - {b^3}} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {bc\left( {c + b} \right) - ab\left( {a + b} \right) + \left( {c - a} \right)\left( {{c^2} + {a^2} + ac} \right)} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {b\left( {{c^2} + cb - {a^2} - ab} \right) + \left( {c - a} \right)\left( {{c^2} + {a^2} + ac} \right)} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {b\left( {{c^2} - {a^2}} \right) + {b^2}\left( {c - a} \right)} \right) + \left( {c - a} \right)\left( {{c^2} + {a^2} + ac} \right)$
$ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {c - a} \right)\left( {b\left( {c + a} \right) + {b^2} + {c^2} + {a^2} + ac} \right)$
Answer $ \Rightarrow \left( {b - c} \right)\left( {a - b} \right)\left( {c - a} \right)\left( {{a^2} + {b^2} + {c^2} + ac + bc + ab} \right)$
Note: In these questions, try to solve the terms one by one by taking common out and using formulas to simplify.
Last updated date: 18th Sep 2023
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