Question

using factor theorem show that (x-y) is a factor of x(y² -z² ) +y(z²-x² )+ z(x²-y²) 

Answer 1

Let x-y is a factor
Then x = y

> x(y² -z² ) +y(z²-x² )+ z(x²-y²)
= x(x²-z²) + x(z²-x²) + z(x²-x²)
= x³ - xz² + xz² - x³ + z(0)
= 0 + 0
= 0

The remainder is zero.
So, x-y is a factor of x(y² -z² ) +y(z²-x²) + z(x²-y²)

Hence proved

Answer 2

Hi ,

Let p( x ) = x(y²-z²)+y(z²-x²)+z(x²-y²)

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If p( y ) = 0 then ( x - y ) is a factor of p( x ) .

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

= y³ - yz² + yz² - y

= 0

Therefore ,

( x - y ) is a factor of p ( x ) .

I hope this helps you.

:)