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 "f(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),h'(x)=2c_(1)x+c_(2)`
`f'(x)=2a_(1),g''(x)=2b_(1),h''(x)=2c_(1),`
`"and "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 values of x, where
`phi(x)=|{:(f(x),g(x),h(x)),(f'(x),g'(x),h'(x)),(f''(x),g''(x),h''(x)):}|`
`therefore" "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+|{:(f(x),g(x),h(x)),(f'(x),g'(x),h'(x)),(0,0,0):}|`
= 0 + 0 + 0 = 0 for all values of x
`therefore" "phi(x)=` constant for all
Hence, `phi(x)` is a constant polynomial.