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 problems: Problem 10: To factorize the expression x^2 - 1 completely, we recognize it as a difference of squares, which follows the form a^2 - b^2 = (a-b)(a+b). Step 1: Identify a and b. In x^2 - 1, we have a^2 = x^2 a = x and b^2 = 1 b = 1. Step 2: Apply the difference of squares formula. x^2 - 1 = (x-1)(x+1) The completely factorized expression is (x-1)(x+1). Problem 11: To write 5 × 5 × 5 × 5 × 5 using indices, we count how many times the base number 5 is multiplied by itself. Step 1: Identify the base number. The base number is 5. Step 2: Count the number of times the base is multiplied. The number 5 is multiplied by itself 5 times. Step 3: Write in index notation. 5 × 5 × 5 × 5 × 5 = 5^5 The expression in index notation is 5^5. Problem 12: To simplify the expression (3b^2)/(2b^2) using laws of indices, we can cancel out common terms. Step 1: Identify common terms in the numerator and denominator. Both the numerator and the denominator have b^2. Step 2: Cancel out the common terms. (3b^2)/(2b^2) = (3 × b^2)/(2 × b^2) = (3)/(2) Alternatively, using the law of indices (a^m)/(a^n) = a^m-n: (3b^2)/(2b^2) = (3)/(2) × b^2-2 = (3)/(2) × b^0 Since any non-zero number raised to the power of 0 is 1 (b^0 = 1): (3)/(2) × 1 = (3)/(2) The simplified expression is (3)/(2). Problem 13: To factorize the expression 4x^2 + 4x, we find the greatest common factor (GCF) of the terms and factor it out. Step 1: Identify the terms in the expression. The terms are 4x^2 and 4x. Step 2: Find the GCF of the terms. The numerical coefficients are 4 and 4, so the common numerical factor is 4. The variable parts are x^2 and x, so the common variable factor is x. The GCF is 4x. Step 3: Factor out the GCF from each term. 4x^2 = 4x × x 4x = 4x × 1 Step 4: Write the expression in factored form. 4x^2 + 4x = 4x(x + 1) The factorized expression is 4x(x+1). Problem 15A: To expand the expression (3x+2)(x-4), we use the FOIL method (First, Outer, Inner, Last). Step 1: Multiply the First terms. (3x)(x) = 3x^2 Step 2: Multiply the Outer terms. (3x)(-4) = -12x Step 3: Multiply the Inner terms. (2)(x) = 2x Step 4: Multiply the Last terms. (2)(-4) = -8 Step 5: Add all the products together. 3x^2 - 12x + 2x - 8 Step 6: Combine like terms. 3x^2 + (-12x + 2x) - 8 = 3x^2 - 10x - 8 The expanded expression is 3x^2 - 10x - 8.