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Question

Using factor theorem, show that $$a - b , b - c$$ and $$c - a$$ are the factors of $$a \left( b ^ { 2 } - c ^ { 2 } \right) + b \left( c ^ { 2 } - a \right) ^ { 2 }$$$$c \left( a ^ { 2 } - b ^ { 2 } \right)$$


Solution

$$a({b^2} - {c^2}) + b({c^2} - {a^2}) + c({a^2} - {b^2})$$

$$ \Rightarrow a{b^2} - a{c^2} + b{c^2} - b{a^2} + c{a^2} - c{b^2}$$

$$ \Rightarrow a{b^2} - b{a^2} - a{c^2} + b{c^2} - c{b^2} + c{a^2}$$

$$ \Rightarrow a{b^2} - b{a^2} + b{c^2} - a{c^2} - c{b^2} + c{a^2}$$

$$ \Rightarrow ab(b - a) + {c^2}(b - a) - c({b^2} - {a^2})$$

$$\Rightarrow ab(b - a) + {c^2}(b - a) - c(b + a)(b - a)$$

$$ \Rightarrow (b - a)\left( {ab + {c^2} - c(b + a)} \right)$$

$$\Rightarrow (b - a)(ab + {c^2} - cb - ac)$$

$$ \Rightarrow (b - a)(ab - bc - ac + {c^2})$$

$$\Rightarrow  (b - a)\left( {b(a - c) - c(a - c)} \right)$$

$$ \Rightarrow (b - a)(a - c)(b - c)$$

$$\Rightarrow (a - b)(b - c)(c - a)$$ Hence proof

Mathematics

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