If `f(x),g(x)a n dh(x)` are three polynomial of degree 2, then prove that `varphi(x)=|f(x)g(x)h(x)f'(x)g'(x h '(x)f' '(x)g' '(x h ' '(x)|` is a constant polynomial.

Explanation

Let `a(x)=a_(1)x^(2)+a_(2)x+a_(3),g(x)=b_(1)x^(2)+b_(2)x+b_(3)`
` " and "h(x) =c_(1)x^(2)+c_(2)x+c_(3)." Then "`
`f(x)=2a_(1)x+a_(2),g(x)=2b_(1)x+b_(2)`
`" and "h(x)=2c_(1)x+c_(2)`
`f(x)=2a_(1),g(x)=2b_(1)h''(x)=2c_(1)`
`f''(x)=g'''(x)=h'''(x)=0`
In order to Prove that `phi (x)` is a constant polynomial it is sufficient to show that `phi (x) =0` for all x. Now
`phi (x)= |{:(f(x),,g(x),,h(x)),(f'(x),,g'(x),,h'(x)),(f''(x),,g''(x),,h''(x)):}|`
`rArr phi (x) =|{:(f'(x),,g'(x),,h'(x)),(f'(x),,g'(x),,h'(x)),(f''(x),,g''(x),,h''(x)):}|`
`+|{:(f(x),,g(x),,h(x)),(f''(x),,g''(x),,h''(x)),(f''(x),,g''(x),,h''(x)):}|`
`+|{:(f(x),,g(x),,h(x)),(f'(x),,g'(x),,h'(x)),(f'''(x),,g'''(x),,h'''(x)):}|`
`=0+0+ |{:(f(x),,g(x),,h(x)),(f'(x),,g'(x),,h'(x)),(0,,0,,0):}|`
`=0+0+0+=0 ` for all x
`rArr phi (x)=` constant for all
Hence `phi(x)` is a constant polynomial.