Using factor theorem, show that x-y, y-z, z-x are the factors of x2(y-z)+y2(z-x)+z2(x-y) - Maths - Polynomials

Question

Using factor theorem, show that x-y, y-z, z-x are the factors of
x2(y-z)+y2(z-x)+z2(x-y)

Ashutosh Verma · Accepted Answer

Solution :

We know Factor theorem " â€‹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 polynomial 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
)â€‹

= 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
getâ€‹

= x2( y - x ) +
y2(x - x ) +
x2( x - y
)â€‹
= x2y - x3 + 0 +
x3 - x2y

= 0

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