Question

using factor theorem prove that (x-y),(y-z),(z-x) are factors of x²(y-z)+y²(z-x)+z²(x-y)​

Answer 1

We know Factor theorem " When x-c is a factor of the polynomial then f(c)=0 . "

So,

We have polynomial = x2( y - z ) + y2(z - x ) + z2( x - y )

So according to factor theorem if ( x - y ) is factor of our polonomial so it gives it value zero at x = y

We substitute x = y , And get

= y2( y - z ) + y2(z - y) + z2( y - y )

= y3 - y2z + y2z - y3 + 0

= 0

So ( x - y ) is a factor of our polynomial.

Now check for ( y - z ) is a factor of our polynomial or not

So we substitute y = z , And get

= x2( z - z ) + z2(z - x ) + z2( x - z )

= 0 + z3 - z2x + z2x - z3

= 0

So ( y - z ) is a factor of our polynomial.

Now check for ( z - x ) is a factor of our polynomial or not.

So we substitute z = x , And get

= x2( y - x ) + y2(x - x ) + x2( x - y )

= x2y - x3 + 0 + x3 - x2y

= 0

So ( z - x ) is a factor of our polynomial

Answer 2

Answer:

put x - y = 0, put y = x

so

x^2(x - z) + x^2(z - x)

x^2(x - z) - x^2(x - z)

= 0

hence x - y is its factor

similarly y - z and z - x are its factor