This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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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 of the variables in the term. Degree = 1+4 = 5. The degree of 3xy^4 is 5. b) 10x^2y^3z^4 Step 1: Identify the exponents of the variables. The exponent of x is 2. The exponent of y is 3. The exponent of z is 4. Step 2: Sum the exponents of the variables in the term. Degree = 2+3+4 = 9. The degree of 10x^2y^3z^4 is 9. c) Q(x) = 7x^6 - 4x^4 + 2x^2 - x + 1 Step 1: Identify the exponent of each term. The exponents are 6, 4, 2, 1, 0 (for the constant term 1 = 1x^0). Step 2: The degree of a polynomial in one variable is the highest exponent of the variable. The highest exponent is 6. The degree of Q(x) 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 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 ((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. 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). Here, 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. 7x^4 (power 4) -2x^2 (power 2) 3x (power 1) 5 (power 0, constant term) Step 2: Arrange the terms from the highest power to the lowest. 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 P(x). P(2) = 2(2)^3 - 5(2)^2 + 4(2) - 1 Step 2: Calculate the value. P(2) = 2(8) - 5(4) + 8 - 1 P(2) = 16 - 20 + 8 - 1 P(2) = -4 + 8 - 1 P(2) = 4 - 1 P(2) = 3 The value of P(2) is 3.