If `f(x)` is a polynomial of degree `<3,` prove that `|1af(a)//(x-a)1bf(b)//(x-b)1cf(c)//(x-c)|-:|1a a^2 1bb^2 1cc^2|=(f(x))/((x-a)(x-b)(x-c))`

If int(tan^(9)x)dx=f(x)+log|cosx|, where f(x) is a polynomial of degree n in tan x, then the value of n is

If g(x)=(f(x))/((x-a)(x-b)(x-c)) ,where f(x) is a polynomial of degree <3 , then intg(x)dx=|[1,a,f(a)log|x-a|],[1,b,f(b)log|x-b|],[1,c,f(c)log|x-c|]|-:|[1,a, a^2],[ 1,b,b^2],[ 1,c,c^2]|+k (dg(x))/(dx)=|[1,a,-f(a)(x-a)^(-2)],[1,b,-f(b)(x-b)^(-2)],[1,c,-f(c)(x-c)^(-2)]|:-|[1,a,a^2],[1,b,b^2],[1,c,c^2]|

If a^2+b^2+c^2=-2a n df(x)= |a+a^2x(1+b^2)x(1+c^2)x(1+a^2)x1+b^2x(1+c^2)x(1+a^2)x(1+b^2)x1+c^2x| , then f(x) is a polynomial of degree 0 b. 1 c. 2 d. 3

p(x) is a polynomial of degree 1 and q(x) is a polynomial of degree 2. What kind of the polynomial p(x) xx q(x) is ?

Let f(x) be a polynomial of degree 5 such that f(|x|)=0 has 8 real distinct , Then number of real roots of f(x)=0 is ________.

Let f(x) be a polynomial of degree 3 such that f(3)=1,f^(prime)(3)=-1,f^('')(3)=0,a n df^(''')(3)=12. Then the value of f^(prime)(1) is (a) 12 (b) 23 (c) -13 (d) none of these

If f(x), h(x) are polynomials of degree 4 and |(f(x), g(x),h(x)),(a, b, c),(p,q,r)| =mx^4+nx^3+rx^2+sx+r be an identity in x, then |(f''(0) - f''(0),g''(0) - g''(0),h''(0) -h''(0)),(a,b,c),(p,q,r)| is

f(x) is polynomial of degree 4 with real coefficients such that f(x)=0 satisfied by x=1, 2, 3 only then f'(1) f'(2) f'(3) is equal to -

If find g are polynomials of derrees m and n respectively, and if h(x) = (f^@ g ) (x), then the degree of h is