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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Here are the solutions to the questions. QUESTION 4: ALGEBRAIC EXPRESSIONS Given the polynomial: -7x^4 + 2x^2 + 11 + 5x^5 - 4x 4.1.1 Step 1: Identify the terms and their powers of x. The terms are 5x^5, -7x^4, 2x^2, -4x (which is -4x^1), and 11 (which is 11x^0). Step 2: Arrange the terms in descending order of their powers. 5x^5 - 7x^4 + 2x^2 - 4x + 11 The polynomial in descending powers of x is 5x^5 - 7x^4 + 2x^2 - 4x + 11 4.1.2 Step 1: Count the number of terms in the polynomial. Terms are separated by addition or subtraction signs. The terms are 5x^5, -7x^4, 2x^2, -4x, and 11. There are 5 terms. There are 5 terms in the polynomial. 4.1.3 Step 1: Locate the term containing x^4. The term is -7x^4. Step 2: Identify the numerical factor multiplying x^4. The coefficient of x^4 is -7. The coefficient of x^4 is -7 4.1.4 Step 1: Identify the highest power of x in the polynomial. The powers of x are 5, 4, 2, 1, 0. The highest power is 5. The degree of the polynomial is 5 4.2 Simplify: 4.2.1 15x + 7y + 25x + 3 + 2y Step 1: Group like terms together. (15x + 25x) + (7y + 2y) + 3 Step 2: Combine the like terms. 40x + 9y + 3 The simplified expression is 40x + 9y + 3 4.2.2 4x^2 + 2x + 4 + 3x^2 - 5x - 1 Step 1: Group like terms together. (4x^2 + 3x^2) + (2x - 5x) + (4 - 1) Step 2: Combine the like terms. 7x^2 - 3x + 3 The simplified expression is 7x^2 - 3x + 3 4.3 Write the following in the normal algebraic way: c2 + d3 Step 1: In algebraic notation, coefficients are typically written before the variables. 2c + 3d The normal algebraic way is 2c + 3d 4.4 Write the following statement in algebraic language: Multiply a number by two and add six to the answer. Step 1: Let the unknown number be represented by a variable, for example, x. Step 2: Translate "multiply a number by two" into an expression. 2x Step 3: Translate "add six to the answer" into an expression. 2x + 6 The algebraic expression is 2x + 6 QUESTION 5: EXPONENTS 5.1 Write each of the following in exponential notation: 5.1.1 512 Step 1: Find the prime factorization of 512. 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 Step 2: Express the number as a base raised to an exponent. 512 = 2^9 In exponential notation, 512 is 2^9 5.1.2 2^3 × 3^3 Step 1: Use the exponent rule (a × b)^m = a^m × b^m. 2^3 × 3^3 = (2 × 3)^3 Step 2: Simplify the base. (2 × 3)^3 = 6^3 Step 3: Calculate the value. 6^3 = 6 × 6 × 6 = 216 In exponential notation, 2^3 × 3^3 is 6^3 (which equals 216). 5.2 Simplify: 5.2.1 2^-4 × 2^5 × 2^0 Step 1: Use the exponent rule a^m × a^n = a^m+n. 2^-4 × 2^5 × 2^0 = 2^(-4) + 5 + 0 Step 2: Simplify the exponent. 2^(-4) + 5 + 0 = 2^1 Step 3: Evaluate the expression. 2^1 = 2 The simplified expression is 2 5.2.2 (-3x)^2 Step 1: Apply the exponent to both the coefficient and the variable inside the parentheses. (-3x)^2 = (-3)^2 × x^2 Step 2: Calculate the square of the coefficient. (-3)^2 = (-3) × (-3) = 9 Step 3: Combine the results. 9x^2 The simplified expression is 9x^2 What's next? Send 'em! 📸