Justify each statement by giving the correct property of absolute value. Assume all variables represent real numbers. See the example in this section.

$$|5+x| \leq|5|+|x|$$

Find each product. See the examples in this section.

$$(r-3 s+t)(2 r-s+t)$$

Match each polynomial in Column I with its factored form in Column II.

$$\begin{array}{ll} \textbf{Column I} \qquad \quad & \textbf{Column II} \\ 25 y^2-x^2 \quad & \text{A. } (x+5 y)(x-5 y) \\ &\text{B. } (x+5 y)^2 \\ &\text{C. } (x-5 y)^2\\ &\text{D. } (5 y+x)(5 y-x)\\ \end{array}$$

Match each polynomial in Column I with its factored form in Column II.

$$\begin{array}{ll} \textbf{Column I} \qquad \quad & \textbf{Column II} \\ x^2-25 y^2 \quad & \text{A. } (x+5 y)(x-5 y) \\ &\text{B. } (x+5 y)^2 \\ &\text{C. } (x-5 y)^2\\ &\text{D. } (5 y+x)(5 y-x)\\ \end{array}$$

Match each polynomial in Column I with its factored form in Column II.

$$\begin{array}{ll} \textbf{Column I} \qquad \quad & \textbf{Column II} \\ x^2-10 x y+25 y^2 \quad & \text{A. } (x+5 y)(x-5 y) \\ &\text{B. } (x+5 y)^2 \\ &\text{C. } (x-5 y)^2\\ &\text{D. } (5 y+x)(5 y-x)\\ \end{array}$$

Match each polynomial in Column I with its factored form in Column II.

$$\begin{array}{ll} \textbf{Column I} \qquad \quad & \textbf{Column II} \\ x^2+10 x y+25 y^2 \quad & \text{A. } (x+5 y)(x-5 y) \\ &\text{B. } (x+5 y)^2 \\ &\text{C. } (x-5 y)^2\\ &\text{D. } (5 y+x)(5 y-x)\\ \end{array}$$

Simplify each expression. Assume all variables represent positive real numbers. See the examples in this section.

$$\sqrt{8 x^5 z^8}$$

Justify each statement by giving the correct property of absolute value. Assume all variables represent real numbers. See the example in this section.

$$|k-m| \leq|k|+|-m|$$

Simplify each expression. See the example in this section.

$$(-4)^{1 / 2}$$

The table indicates the number of Internet users in North America (in millions).

Number of Internet Users in North America

The polynomial

$$.146 x^4-2.54 x^3+11.0 x^2+16.6 x+51.5$$

models the number of users in year $x$, where $x=0$ corresponds to $1997$, $x=1$ corresponds to $1998$, and so on. For each given year,

(a ) use the table to determine the number of users, and then (b ) use the polynomial to determine the number of users. (c ) How closely does the polynomial approximate the number of users compared with the data?

$$2002$$

Perform each addition or subtraction. See the example in this section.

$$\dfrac{1}{a}-\dfrac{b}{a^2}$$

Find each product. See the examples in this section.

$$(m-n+k)(m+2 n-3 k)$$

Evaluate each expression if $p=-4, q=8$, and $r=-10$. See the mentioned example.

$$\dfrac{3 q}{r}-\dfrac{5}{p}$$

Find each product. See the examples in this section.

$$(k+2)\left(12 k^3-3 k^2+k+1\right)$$