The sum of the coefficients in the expansion of $\left(1+\frac{x}{2}\right)^{12}$ is

If the coefficients of $x^2$ and $x^3$ are both zero in the expansion of the expression $(1 + ax + bx^2)(1 - 3x)^{15}$ in powers of $x$,then the ordered pair $(a, b)$ is equal to

By neglecting $x^4$ and higher powers of $x$, find the approximate value of $\sqrt[3]{x^2+64}-\sqrt[3]{x^2+27}$.

For $|x| < 1$,the constant term in the expansion of $\frac{1}{(x-1)^2(x-2)}$ is

Expand the expression $\left(x+\frac{1}{x}\right)^{6}$.

Let $n > 2$ be an integer and define a polynomial $p(x) = x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$,where $a_0, a_1, \ldots, a_{n-1}$ are integers. Suppose we know that $n p(x) = (1 + x) p'(x)$. If $b = p(1)$,then: