Question

Verify the following: (a^2-b^2)(a^2+b^2)+(b^2-c^2)(b^2+c^2)+(c^2-a^2)(c^2+a^2)=0
Use factorisation method

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(

c

d

), 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 solve 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)(∵(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 (a−b) is a factor of a(b

2

−c

2

)+b(c

2

−a

2

)+c(a

2

−b

2

).