If `f(x),g(x)andh(x)` are three polynomials of degree 2 , then prove that `phi(x)=abs{:(f(x)" "g(x)" "h(x)),(f'(x)" "g'(x)" "h'(x)),(f''(x)" "g''(x)" "h''(x)):}` is a constant polynomial.

Options

A. 2
B. 3
C. 4
D. none of these

Answer

D

Explanation

Since, `f(x), g(x) and h(x)` are polynomials of degree 3.
`:. f'''(x) = g'''(x) = h'''(x) = 0`
Now, `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))|`
`rArr phi' (x) = 0 + 0 + |(f(x),g(x),h(x)),(f'(x),g'(x),h'(x)),(0,0,0)| = 0`
`:. phi(x)` = constant
Hence, `phi(x)`is a constant polynomial