What is the degree of each polynomial and which expressions are polynomials with reasons?

The exponent of $x$ is 1.
The exponent of $y$ is 4.

Degree $= 1+4 = 5$.
The degree of $3xy^4$ is $\boxed{5}$.

b) $10x^2y^3z^4$

The exponent of $x$ is 2.
The exponent of $y$ is 3.
The exponent of $z$ is 4.

Degree $= 2+3+4 = 9$.
The degree of $10x^2y^3z^4$ is $\boxed{9}$.

c) $Q(x) = 7x^6 - 4x^4 + 2x^2 - x + 1$

The exponents are $6, 4, 2, 1, 0$ (for the constant term $1 = 1x^0$).

The highest exponent is 6.
The degree of $Q(x)$ is $\boxed{6}$.

2. Determine which of the following expressions are polynomials. Give reasons for your answer.

a) $3x^3 - 2x + 7$
This is a polynomial.
Reason: All exponents of the variable $x$ are non-negative integers ($3, 1, 0$).

b) $\frac{1}{x} + 4$
This is not a polynomial.
Reason: The term $\frac{1}{x}$ can be written as $x^{-1}$, which has a negative exponent ($-1$). Polynomials require non-negative integer exponents.

c) $5x^{1/2} + x$
This is not a polynomial.
Reason: The term $5x^{1/2}$ involves a fractional exponent ($\frac{1}{2}$). Polynomials require integer exponents.

3. Find the remainder when the polynomial $P(x) = 2x^3 - 5x^2 + 4x - 3$ is divided by $x-2$.

The Remainder Theorem states that if a polynomial $P(x)$ is divided by $x-c$, the remainder is $P(c)$.
Here, $P(x) = 2x^3 - 5x^2 + 4x - 3$ and the divisor is $x-2$, so $c=2$.

The remainder is $\boxed{1}$.

4. Rewrite the polynomial in descending powers of $x$: $3x + 7x^4 - 2x^2 + 5$.

$7x^4$ (power 4)
$-2x^2$ (power 2)
$3x$ (power 1)
5 (power 0, constant term)

The polynomial in descending powers of $x$ is $\boxed{7x^4 - 2x^2 + 3x + 5}$.

5. Given $P(x) = 2x^3 - 5x^2 + 4x - 1$, find the value of $P(2)$.

The value of $P(2)$ is $\boxed{3}$.