If `p(x), q(x)` and `r(x)` are three polynomials of degree 2, then prove that `|(p(x), q(x), r(x)), (p'(x), q'(x), r'(x)), (p''(x), q''(x), r''(x))|` is independent of x.

If p(x) ,q(x) and r(x) and three polynomials of degree 2, then |{:(p(x),q(x),r(x)),(p'(x),q'(x),r'(x)),(p''(x),q''(x),r''(x)):}| is ………….of x .

Let p(x)=(x-1)(x-2)(x-3) for how many polynomials R(x) of degree 3 such that P(Q(x))=P(x)*R(x)?

|(x,p,q),(p,x,q),(p,q,x)| =

Prove that: |{:(a, b, c), (x, y, z), (p, q, r):}|=|{:(y, b, q), (x, a, p), (z, c, r):}|

Assertion : 3x^(2)+x-1=(x+1)(3x-2)+1 . Reason : If p(x) and g(x) are two polynomials such that degree of p(x)ge degree of g(x) and g(x)ge0 then we can find polynomials q(x) and r(x) such that p(x)=g(x).q(x)+r(x) , where r(x)=0 or degree of r(x)lt degree of g(x) .

The polynomial p(x) is such that for any polynomial q(x) we have p(q(x))=q(p(x)) then p(x) is

Give an example of polynomials f(x),g(x),q(x) and r(x) satisfying f(x)=g(x)dot q(x)+r(x), where degree r(x)=0