Polynomials

In the expression $(x+1)(x+4)(x+9)(x+16)\cdots(x+400)$ the coefficient of $x^{19}$ is

If $f(x)$ is a polynomial satisfying $f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ and $f(3)=28$, then $f(4)$ is

If $2x^4 + x^3 - 11x^2 + x + 2 = 0$ then the values of $x + \dfrac{1}{x}$ are:

Find $k$ in the equation $x^3 - 6x^2 + kx + 64 = 0$ if roots are in geometric progression.

If $(1 + x - 2x^2)^6 = 1 + a_1 x + a_2 x^2 + \ldots + a_{12} x^{12}$, then the value of $a_2 + a_4 + a_6 + \ldots + a_{12}$ is:

Let $f(x)$ be a polynomial function of second degree and $f(1) = f(–1)$. If $a, b, c$ are in A.P. then $f'(a), f'(b), f'(c)$ are in.

Let f(x) be a polynomial of degree four, having extreme value at x = 1 and x = 2. If $\lim _{{x}\rightarrow0}[1+\frac{f(x)}{{x}^2}]=3$, then f(2) is

If (1 - x + x² )ⁿ = a + a₁x + a₂x² + ... + a_2nx²ⁿ , then a₀ + a₂ + a₄ + ... + a_2n is

The equation (x-a)³+(x-b)³+(x-c)³ = 0 has

α, β are the roots of the an equation $x^2- 2x cosθ + 1 = 0$, then the equation having roots αn and βn is

If (1 + x – 2x²)⁶ = 1 + a₁x + a₂x² + ... + a₁₂x¹², then the value a₂ + a₄ + a₆ + ... + a₁₂

If a + b + c = 0, then the value of $\frac{a^2}{bc}+\frac{b^2}{ca}+\frac{c^2}{ab}$

Roots of equation are $ax^2-2bx+c=0$ are n and m , then the value of $\frac{b}{an^2+c}+\frac{b}{am^2+c}$ is

For what value of p, the polynomial  $x^4-3x^3+2px^2-6$ is exactly divisible by $(x-1)$