Determine if each of the following statements are true or false. Provide a brief explanation to justify each of your answers. a. The roots of a polynomial function are rational numbers. b. The process of finding zeros of a polynomial function is also referenced as finding the roots of the polynomial function. c. We can apply the zero-product property to determine the zeros of a polynomial function in factored form. d. The function used to model the box problem in Module 3 has two roots, at x = 0 and x = 4.25. e. The zeros of any function f represent the value(s) of x that when input into the function f return a value of 1 for the output variable f(x). f. The graph of a polynomial function is always a smooth curve, g. Polynomial functions are not always continuous.

Find the distance between the points.

$$(-4.2,3.1),(-12.5,4.8)$$

the distance east or west from the prime meridian

Write always, sometimes, or never to complete a true statement.

A sequence of a rotation and a dilation will $\underline{\text{?}}$ result in an image that is the same size and orientation as the preimage.

Is the elastic limit a limiting force or a limiting distance?

Suppose G is an infinite cyclic group with generator $\sigma$. (a) Prove that multiplication by

$$\sigma-1 \in \mathbb{Z} G$$

defines a free G-module resolution of

$$\mathbb{Z}: 0 \rightarrow \mathbb{Z} G \stackrel{\sigma-1}{\rightarrow} \mathbb{Z} G \rightarrow \mathbb{Z} \rightarrow 0.$$

(b) Show that

$$H^{0}(G, A) \cong A^{G},$$

that

$$H^{1}(G, A) \cong A /(\sigma-1) A.$$

and that $H^{n}(G, A)=0$ for all n $\geq$ 2. Deduce that

$$H^{1}(G, \mathbb{Z} G) \cong \mathbb{Z}$$

(so free modules neednot be cohomologically trivial).

Find the distance between the points.

$$(6.2,5.4),(-3.7,1.8)$$

Scientists think that the global population of tigers is falling exponentially. Estimates suggest that in 1970 there were 37 000 tigers but by 1980 the number had dropped to 22 000. (a) The number T of tigers n years after 1970 can be modelled by

$$T = k a ^ { n }$$

. (i) Write down the value of k. (ii) Show that a = 0.949 to three significant figures. (b) What does the model predict that the population will be in 2020? (c) When the population reaches 1000, the tiger population will be described as ‘near extinction’. In which year will this happen? In the year 2000 a worldwide ban on the sale of tiger products was implemented, and it is believed that by 2010 the population of tigers had recovered to 10 000. (d) If the recovery has been exponential, find a model of the form

$$T = k a ^ { m }$$

connecting the number of tigers (T) with the number of years after 2000 (m). (e) If each year since 2000 the rate of growth has been the same, find the percentage increase in the tiger population each year.

Find the distance from $(-1,7)$ to the line $x=3$.

Make sense of problems. A sale on dog food that costs $10 a bag says buy 1, get one$$$\frac { 1 } { 2 }$ off. What is the true percent discount on the dog food?

In 1975, an airplane carrying an atomic clock flew back and forth for $15 \mathrm{~h}$ at an average speed of $140 \mathrm{~m} / \mathrm{s}$ as part of a time-dilation experiment. The time on the clock was compared to the time on an atomic clock kept on the ground. How much time did the airborne clock "lose" with respect to the clock on the ground?

In previous question, let the height h equal $6.00$ m and the speed v$_{boy}$ equal $2.00$ m/s. Assume that the food pack starts from rest.

(a) Tabulate and graph the speed–time graph.

(b) Tabulate and graph the acceleration–time graph. Let the range of time be from $0$ s to $5.00$ s and the time intervals be $0.500$ s.

The distance from the surface of a lens to the focal point is the

Refer to the table. Which lap has the slowest time?