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Question

Using factor theorem, show that $$a-b$$ is a factor of $$a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)$$


Solution

We know that the factor theorem states that if the polynomial $$p(x)$$ is divided by $$(cx−d)$$ and the remainder, given by $$p\left( \dfrac { d }{ c }  \right)$$, is equal to zero, then $$(cx−d)$$ is a factor of $$p(x)$$.

Consider the given expression $$a(b^2−c^2)+b(c^2−a^2)+c(a^2−b^2)$$ and solving it as follows:

$$a(b^{ 2 }−c^{ 2 })+b(c^{ 2 }−a^{ 2 })+c(a^{ 2 }−b^{ 2 })\\ =ab^{ 2 }−ac^{ 2 }+bc^{ 2 }−ba^{ 2 }+c(a-b)(a+b)\quad \quad \quad \quad \quad \quad (\because \quad (x+y)(x-y)=x^{ 2 }-y^{ 2 })\\ =ab^{ 2 }−ba^{ 2 }−ac^{ 2 }+bc^{ 2 }+c(a-b)(a+b)\quad \quad \quad \quad \quad \\ =ab(b-a)-(a-b)c^{ 2 }+c(a-b)(a+b)\\ =-ab(a-b)-(a-b)c^{ 2 }+c(a-b)(a+b)\\ =(a-b)(-ab-c^{ 2 }+c(a+b))\\ =(a-b)(c(a+b)-ab-c^{ 2 })$$

Hence, by factor theorem we have proved that $$(a-b)$$ is a factor of $$a(b^2−c^2)+b(c^2−a^2)+c(a^2−b^2)$$.

Mathematics

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