Find the degree of each polynomial

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 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 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 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) (1)/(x)+4 This is not a polynomial. Reason: The term (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 ((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 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 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 3.