Here are the solutions to the problems. 1. Find the degree of each polynomial a) $3xy^4$ Step 1: Identify the exponents of the variables. The exponent of $x$ is 1. The exponent of $y$ is 4. Step 2: Sum the exponents. Degree $= 1 + 4 = 5$. The degree of the polynomial is $\boxed{5}$. b) $10x^2y^3+z^4$ Step 1: Find the degree of each term. For $10x^2y^3$: sum of exponents is $2+3=5$. For $z^4$: exponent is $4$. Step 2: The degree of the polynomial is the highest degree among its terms. The highest degree is $5$. The degree of the polynomial is $\boxed{5}$. c) $Q(x)=7x^6-4x^4+2x^3-x+1$ Step 1: Identify the highest exponent of the variable $x$. The exponents are $6, 4, 3, 1, 0$ (for the constant term). Step 2: The highest exponent is $6$. 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 $x-2$. Step 1: Apply the Remainder Theorem. 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 $c=2$. Step 2: Evaluate $P(2)$. $$ P(2) = 2(2)^3 - 5(2)^2 + 4(2) - 3 $$ $$ P(2) = 2(8) - 5(4) + 8 - 3 $$ $$ P(2) = 16 - 20 + 8 - 3 $$ $$ P(2) = -4 + 8 - 3 $$ $$ P(2) = 4 - 3 $$ $$ P(2) = 1 $$ The remainder is $\boxed{1}$. 4. Rewrite the polynomial in descending powers of $x: 3x+7x^4-2x^2+5$. Step 1: Identify the terms and their powers of $x$. $3x$ has power $1$. $7x^4$ has power $4$. $-2x^2$ has power $2$. $5$ is a constant term, which can be considered $5x^0$. Step 2: Arrange the terms from the highest power of $x$ to the lowest. The order of powers should be $4, 2, 1, 0$. $$ 7x^4 - 2x^2 + 3x + 5 $$ 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 $P(2)$. Step 1: Substitute $x=2$ into the polynomial expression. $$ P(2) = 2(2)^3 - 5(2)^2 + 4(2) - 1 $$ Step 2: Evaluate each term. $$ P(2) = 2(8) - 5(4) + 8 - 1 $$ $$ P(2) = 16 - 20 + 8 - 1 $$ Step 3: Perform the additions and subtractions. $$ P(2) = -4 + 8 - 1 $$ $$ P(2) = 4 - 1 $$ $$ P(2) = 3 $$ The value of $P(2)$ is $\boxed{3}$.