Chapter 1: Fundamentals

Rationalize the denominator.

$\frac{3}{2-\sqrt{5}}$

Find the number of real solutions of the equation $x^2+rx-s=0\left(s>0\right)$by using discriminant.

The bar graph shows the daily high temperatures for Omak, Washington, and Geneseo, New York, during a certain week in June. Let $T_{\mathrm{O}}$represent the temperature in Omak and $T_{\mathrm{G}}$the temperature in Geneseo. Calculate and $T_{\mathrm{O}}-T_{\mathrm{G}}$and $\left|T_{\mathrm{O}}-T_{\mathrm{G}}\right|$for each day shown. Which of these two values gives more information?

DISCUSS: Power of i Calculate the first 12 powers of i, that is, $i,i^2,i^3,\mathrm{...},i^{12}$. Do you notice a pattern? Explain how you would calculate any whole number power of i, using the pattern that you have discovered. Use this procedure to calculate $i^{4446}$.

Write each number in decimal notation.

1. $7.1\times {10}^{14}$
2. $6\times {10}^{12}$
3. $8.55\times {10}^{-3}$
4. $6.257\times {10}^{-10}$

Factor the expression by grouping terms

$3x^3-x^2+6x-2$

A boardwalk is parallel to and 210 ft inland from a straight shoreline. A sandy beach lies between the boardwalk and the shoreline. A man is standing on the boardwalk, exactly 750 ft across the sand from his beach umbrella, which is right at the shoreline. The man walks 4 ft/s on the boardwalk and 2 ft/s on the sand. How far should he walk on the boardwalk before veering off onto the sand if he wishes to reach his umbrella in exactly 4 min 45 s?

Solve the absolute value Inequality. Express the answer using interval notation and graph the solution set.

$\left|\frac{x+1}{2}\right|\ge 4$

Tangent Line to a Circle

(a)Find an equation for the line tangent to the$x^2+y^2=25$ at the point$(3,-4).$ (See the figure)

(b)At what other point on the circle will a tangent line be parallel to the tangent line in part (a)?

Find the center and radius of the circle, and sketch its graph.

${\left(x+1\right)}^2+y^2=9$