Using factor theorem, show that a−b is a factor of a(b2−c2)+b(c2−a2)+c(a2−b2)
We know that the factor theorem states that if the polynomial p(x) is divided by (cx−d) and the remainder, given by p(dc), is equal to zero, then (cx−d) is a factor of p(x).
Consider the given expression a(b2−c2)+b(c2−a2)+c(a2−b2) and solving it as follows:
a(b2−c2)+b(c2−a2)+c(a2−b2)=ab2−ac2+bc2−ba2+c(a−b)(a+b)(∵(x+y)(x−y)=x2−y2)=ab2−ba2−ac2+bc2+c(a−b)(a+b)=ab(b−a)−(a−b)c2+c(a−b)(a+b)=−ab(a−b)−(a−b)c2+c(a−b)(a+b)=(a−b)(−ab−c2+c(a+b))=(a−b)(c(a+b)−ab−c2)
Hence, by factor theorem we have proved that (a−b) is a factor of a(b2−c2)+b(c2−a2)+c(a2−b2).