Prove that if F is a subfield of a field E, and $c \in E,$ then $\theta: F[x] \rightarrow E$ defined by $\theta(f(x))= f(c)$ is a homomorphism.

Find an SVD of the indicated matrix.

$$A = \left[ \begin{array} { r r } { 0 } & { - 2 } \\ { - 3 } & { 0 } \end{array} \right]$$

Lana says there are $4 ^ { 3 }$ eggs in a dozen. Patrick says there are $3 ^ { 4 }$ eggs in a dozen. Is either person correct? Explain your reasoning.

Perform the indicated operations. Where possible, reduce the answer to its lowest terms.

$$\frac { 1 } { 3 } - 5 \left( \frac { 1 } { 2 } + 1 \right)$$

Write the fraction or mixed number as a decimal. 10 4/9

Fill in the blank. One _______ is the measure of a central angle that intercepts an arc equal in length to the radius of the circle.

Name the property of real numbers illustrated by each equation. $\frac{2}{5}+\frac{27}{5} \cdot \frac{5}{27}=\frac{2}{5}+1$

A delivery man starts at the post office, drives 40 km north, then 20 km west, then 60 km northeast, and finally 50 km north to stop for lunch. Use the analytical method to determine the following: (a) Find his net displacement vector. (b) How far is the restaurant from the post office? (c) If he returns directly from the restaurant to the post office, what is his displacement vector on the return trip? (d) What is his compass heading on the return trip? Assume the +x-axis is to the east.

Test for symmetry with respect to the line $\theta=\pi / 2$, the polar axis, and the pole. $r=4 \csc \theta \cos \theta$

What is the partial pressure of hydrogen gas in a mixture of hydrogen and helium if the total pressure is 600 mmHg and the partial pressure of helium is 439 mmHg?

Find and simplify the function values. f(x, y) = xy (b)(-1, 4)

(a) Find the derivative of x lnx. (b) Hence find

$$\int \ln x \ d x$$

.

Calculate the kinetic energy that the earth has because of its rotation about its own axis. Assume that the earth is a uniform sphere and that its path around the sun is circular. For comparison, the total energy used in the United States in one year is about

$$9.3 \times 10 ^ { 19 } \mathrm { J }$$

What changes in physical properties can be used as evidence that a chemical reaction has occurred?