When multiplying, why is the degree of the product different from the degree of the factors?

How does multiplying polynomials compare to multiplying integers?

Justify the following statement: The set of polynomials is closed under addition, subtraction, and multiplication, but not under division.

Determine the open intervals on which the graph of the function is concave upward or concave downward. $f(x)=-3 x^{2}$

Why is it important that the product of two polynomials have only whole number exponents?

The polynomial $-16t^2 + vt + h_0$ models the height, in feet, of a ball $t$ seconds after it is thrown. In this polynomial, $v$ represents the initial vertical velocity of the ball and $h_0$ represents the initial height of the ball.

Min throws a baseball from a height of 6 feet with an initial vertical velocity of 30 feet per second. Which polynomial models the height of the ball, in feet?

Explain what you would do first to simplify the expression below. Justify why and then state the result of performing this step. The expression to simplify is ((2r^2t)^3/(4t^2))^2.

find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix.

$$\left[\begin{array}{ll} {1} & {2} \\ {2} & {1} \end{array}\right]$$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $y=(4 x-3)\left(x^{3}-2 x^{2}\right)$

Explain why the product of two binomials in the form (a + b) (a - b) is a binomial instead of a trinomial.

Polymorphism implies that each different MHC protein binds a different peptide motif. For the MHC class I polymorphisms, how many different MHC proteins are expressed in an individual? How many by the entire human population?

A chemist wishes to estimate the yield $Y$ in a certain process that depends on the temperature $T$ and the pressure $P.$ A linear model for the process is assumed: $Y=a+b_{1} T+b_{2} P,$ where $a, b_{1},$ and $b_{2}$ are unknown parameters. Assume that the observations are as follows.

$$\begin{array}{cccc} \hline Y & {T} & {P} \\ \hline Y_{1} & {T_{1}} & {P_{1}} \\ {Y_{2}} & {T_{2}} & {P_{2}} \\ {\vdots} & {\vdots} & {\vdots} \\ {Y_{n}} & {T_{n}} & {P_{n}} \\ \hline \end{array}$$

Derive equations for the coefficients of the least squares plane, generalizing the technique of the section for the least squares line.

Troy knows that division and multiplication are inverse operations. In other words, multiplication undoes division and division undoes multiplication. You can use multiplication to check an answer to a division problem. Troy challenged Phillip to the matching game, "Divide & Conquer." He said to Phillip, "I'll ask you a division problem. You solve it and turn it around with a multiplication sentence to prove your answer." When Troy said, "3 pies divided in eighths results in 24 pieces," Phillip responded, "If I eat $\frac{1}{8}$ of a pie, 24 times, I've eaten 3 whole pies. $\frac{1}{8} \cdot 24=3.$" State each problem below as a division problem. Then solve the problem and confirm your solution by writing and stating the appropriate multiplication sentence. a. If each box holds 5 books, how many boxes or partial boxes would be filled by 14 books? b. How much does each person get if $\frac{1}{2}$ pound of chocolate is shared equally between 3 people?

Let $S=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}\right\}$ be a set of linearly dependent vectors in V, and let T be a linear transformation from V into V. Prove that the set $\left\{T\left(\mathbf{v}_{1}\right), T\left(\mathbf{v}_{2}\right), \ldots T\left(\mathbf{v}_{n}\right)\right\}$ is linearly dependent.