Factor the polynomial.

$$p^3+216$$

Factor completely.

$$x^3-y^6$$

Factor: $2 x^6+16$

X-rays of wavelength $\lambda=0.140 \mathrm{~nm}$ are scattered from carbon. What is the expected Compton wavelength shift for photons detected at angles (relative to the incident beam) of exactly $45^{\circ}$ ?

Suppose that $X$ and $Y$ are independent random variables, where $X$ is normally distributed with mean 45 and standard deviation 0.5 and $Y$ is normally distributed with mean 20 and standard deviation 0.1. Find $P(40 \leqslant X \leqslant 50, 20 \leqslant Y \leqslant 25)$.

Find the derivative of the function in part (a). Then find the derivative of the function in part (b) or show that the function in part (b) is the same function as that in part (a).

(a) $y=\ln (x-1)+\ln (2 x+1)$\

(b) $y=\ln [(x-1)(2 x+1)]$

If $x$ is normally distributed with $\mu=20.0$ and $\sigma=4.0$, determine the following:

c. $P(x \leq 12)$

Factor.

$$5x^2 - 26x + 5$$

Factor.

$$18x^2 - 12x + 2$$

A photon with a wavelength of $0.1800 \mathrm{~nm}$ is Compton-scattered through an angle of $180^{\circ}$, a) Find the wavelength of the scattered photon? b) How much energy is given to the electron? c) Find the recoil speed of the electron? Is it necessary to use the relativistic kinetic-energy relationship?

Factor.

$$4 y^8-4 y^5$$

Factor.

$$x^2-8 x+15$$

Factor.

$$x^3 y^3+64$$

What is the row space of a matrix? The column space? The null space?