Verify that each equation is an identity. $\frac{\sin ^{2} x}{2-2 \cos x}=\cos ^{2} \frac{x}{2}$

Find $d^2 y / d x^2$ in terms of $x$ and $y$. \

$$x^2+y^2=36$$

This exercise contains polynomials in several variables. Factor the polynomial completely and check using multiplication.

$$-36 x^3 y+62 x^2 y^2-12 x y^3$$

Subtract the two polynomial from each other. $3 x^3-2 x^2 y+x y^2$ , $x^3+x^2 y-x y^2$

Is the algebraic expression a polynomial? If it is, write the polynomial in standard form.

$$2 x + 3 x ^ { 2 } - 5$$

Simplify the expression. $\frac{\frac{z}{1-\frac{z}{2+2 z}}-2 z}{\frac{2 z}{5 z-2}-3}$

Evaluate given integral.
$\int \frac{x^2+2 x}{(x+1)\left(x^2+2 x+2\right)}$ dx

True or False. $3 x^{3}-2 x^{2}-6 x+4=(3 x-2)\left(x^{2}+2\right)$

Explain why the sum of two polynomials will always be a polynomial.

There are 24 male and 36 female patrons in a restaurant. Of the 11 patrons under 10 years old, 6 are male. Of the 14 patrons over 55 years old, 9 are female. A patron is selected at random. Determine whether the events are mutually exclusive or not mutually exclusive. Then probability of $P$ (female or under 10 )

Evaluate the expression for the given values.

$$\sqrt { \left( x _ { 2 } - x _ { 1 } \right) ^ { 2 } + \left( y _ { 2 } - y _ { 1 } \right) ^ { 2 } }$$

for

$$x _ { 1 } = - 2 , x _ { 2 } = 4 , y _ { 1 } = 4 , y _ { 2 } = - 4$$

Multiply and simplify:

$$\left(4 u^2-5 v^2\right)\left(2 u^2+3 v^2\right)$$

Solve each of the following by two methods

$$\begin{array}{l}{\text { (a) }\left(x^{2}+2 y^{2}\right) d x+\left(4 x y-y^{2}\right) d y=0} \\ {\text { (b) }\left(2 x^{2}+2 x y+y^{2}\right) d x+\left(x^{2}+2 x y\right) d y=0}\end{array}$$

The Laguerre polynomials $L_n(x)$ are polynomial solutions to Laguerre's equation

$$x y^{\prime \prime}+(1-x) y^{\prime}+n y=0 .$$

The Laguerre polynomials satisfy Rodrigues's formula,

$$L_n(x)=\frac{e^x}{n !} \frac{d^n}{d x^n}\left(x^n e^{-x}\right) .$$

Use this formula to determine the first four Laguerre polynomials.