Prove that `x^(3p) + x^(3q+1) + x^(3r+2)` is exactly divisible by `x^2+x+1`, if p,q,r is integer.

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

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.Suppose that f(x) = P(x^3) + x Q(x^3) is divisible by x^2 + x+1, then (a) P(x) is divisible by (x-1),but Q(x) is not divisible by x -1 (b) Q(x) is divisible by (x-1), but P(x) is not divisible by x-1 (c) Both P(x) and Q(x) are divisible by x-1 (d) f(x) is divisible by x-1

Find the values of p\ a n d\ q so that x^4+p x^3+2x^2-3x+q is divisible by (x^2-1)

The polynomial px^(3)+4x^(2)-3x+q completely divisible by x^2-1: find the values of p and q. Also for these values of p and q factorize the given polynomial completely.

Statement-1: The coefficient of a^(3)b^(4)c^(3) in the expansion of (a-b+c)^(10) is (10!)/(3!4!3!) Statement-2: The coefficient of x^(p)y^(q)z^(r) in the expansion of (x+y+z)^(n) is (n!)/(p!q!r!) for all integer n.

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

If p(x)=x^(5)+4x^(4)-3x^(2)+1 " and" g(x)=x^(2)+2 , then divide p(x) by g(x) and find quotient q(x) and remainder r(x).

In Figure 1, y = sqrt(3)x and z = 2x . What is the ratio p : q : r in figure 2?

Given that q^2-p r ,0,\ p >0, then the value of |p q p x+q y q r q x+r y p x+q y q x+r y0| is- a. zero b. positive c. negative \ d. q^2+p r