Question

using factor therom show that (a-b)is a factor of a(b^2-c^2+b(c^2-a^2)+c(a^2-b^2)​

Answer 1

Answer:

We know that the factor theorem states that if the polynomial p(x) is divided by (cx−d) and the remainder, given by p( cd ), 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 +bc2 −ba 2+c(a−b)(a+b)(∵(x+y)(x−y)=x 2−y 2 )

=ab 2 −ba 2−ac 2+bc 2+c(a−b)(a+b)

=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(a2−b2 ).

Step-by-step explanation:

hope it helps you