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 2 2.1. Factorize the following completely: 2.1.1. 5x^3y - 10x^2y^2 + 15xy^3 Step 1: Find the greatest common factor (GCF) of all terms. The GCF of the coefficients 5, -10, 15 is 5. The GCF of the x terms x^3, x^2, x is x. The GCF of the y terms y, y^2, y^3 is y. Thus, the GCF of the entire expression is 5xy. Step 2: Factor out the GCF from each term. 5x^3y - 10x^2y^2 + 15xy^3 = 5xy(x^2 - 2xy + 3y^2) The completely factorized expression is 5xy(x^2 - 2xy + 3y^2). 2.1.2. x^3 - 3x^2 - 4x + 12 Step 1: Group the terms. (x^3 - 3x^2) - (4x - 12) Step 2: Factor out the common factor from each group. x^2(x - 3) - 4(x - 3) Step 3: Factor out the common binomial factor (x-3). (x - 3)(x^2 - 4) Step 4: Factor the difference of squares x^2 - 4 as (x-2)(x+2). (x - 3)(x - 2)(x + 2) The completely factorized expression is (x - 3)(x - 2)(x + 2). 2.2. Simplify the following: 2.2.1. ((x^2-1))/(x+1) - ((x-1)(x+1))/(x+2) Step 1: Factor the numerator of the first fraction using the difference of squares formula, a^2-b^2=(a-b)(a+b). ((x-1)(x+1))/(x+1) - ((x-1)(x+1))/(x+2) Step 2: Simplify the first fraction by canceling the common term (x+1), assuming x ≠ -1. (x-1) - ((x-1)(x+1))/(x+2) Step 3: Find a common denominator, which is (x+2). ((x-1)(x+2))/(x+2) - ((x-1)(x+1))/(x+2) Step 4: Combine the numerators over the common denominator. ((x-1)(x+2) - (x-1)(x+1))/(x+2) Step 5: Factor out (x-1) from the numerator. ((x-1)[(x+2) - (x+1)])/(x+2) Step 6: Simplify the expression inside the square brackets. ((x-1)[x+2 - x-1])/(x+2) ((x-1)(1))/(x+2) The simplified expression is (x-1)/(x+2). 2.2.2. (x-sqrt(2))(sqrt(x)+2x^2) Step 1: Use the FOIL method to multiply the two binomials. (x-sqrt(2))(sqrt(x)+2x^2) = x(sqrt(x)) + x(2x^2) - sqrt(2)(sqrt(x)) - sqrt(2)(2x^2) Step 2: Simplify each term. xsqrt(x) = x^1x^1/2 = x^3/2 x(2x^2) = 2x^3 -sqrt(2)sqrt(x) = -sqrt(2x) -sqrt(2)(2x^2) = -2sqrt(2)x^2 Step 3: Combine the simplified terms. x^3/2 + 2x^3 - sqrt(2x) - 2sqrt(2)x^2