Find the degree of each polynomial

Degree $= 1 + 4 = 5$.  
The degree of the polynomial is $\boxed{5}$.

• b) $10x^2y^3+z^4$

The highest degree is 5.  
The degree of the polynomial is $\boxed{5}$.

• c) $Q(x)=7x^6-4x^4+2x^3-x+1$

The degree of the polynomial 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 must have only non-negative integer exponents.

• c) $5x^{1/2}+x$ This is not a polynomial. Reason: The term $5x^{1/2}$ has a fractional exponent ($\frac{1}{2}$). Polynomials must have only non-negative 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 $P(c)$.
In this case, $P(x) = 2x^3-5x^2+4x-3$ and the divisor is $x-2$, so $c=2$.

The remainder is $\boxed{1}$.

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

$3x$ has power 1.
$7x^4$ has power 4.
$-2x^2$ has power 2.
5 is a constant term, which can be considered $5x^0$.

The order of powers should be $4, 2, 1, 0$.

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 $\boxed{3}$.