Show that the polynomial `x^(4p)+x^(4q+1)+x^(4r+2)+x^(4s+3)` is divisible by `x^3+x^2+x+1, w h e r ep ,q ,r ,s in ndot`

Let P(x) and Q(x) be two polynomials. If f(x) = P(x^4) + xQ(x^4) is divisible by x^2 + 1 , then

The sum of the polynomials p(x) =x^(3) -x^(2) -2, q(x) =x^(2) -3x+ 1

Find the number of polynomials of the form x^3+a x^2+b x+c that are divisible by x^2+1,w h e r ea , b ,c in {1,2,3,9,10}dot

The sum of the polynomials p(x) = x^(3) - x^(2) - 2, q(x) = x^(2) - 3x + 1

Prove that p x^(q-r)+q x^(r-p)+r x^(p-q)> p+q+r ,w h e r ep ,q ,r are distinct and x!=1.

Integrate the following w.r.t. x. (x)/(1+x^(4))

Show that the function f given by f(x) = x^(3)-3x^(2)+4x,x in R is increasing on R.

Let the polynomials be (1) -13q^(5) + 4q^(2) + 12q (2) (x^(2) + 4 ) ( x^(2) + 9) (3) 4q^(8) - q^(6) + q^(2) (4) - ( 5)/( 7) y^(12) + y^(3) + y^(5) Then ascending order of their degree is

Why are 1/4 and -1 zeroes of the polynomials p(x)=4x^(2)+3x-1 ?