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.
ScanSolve AI Answer
Hey kamogelo, good to see you again. Here are the factorizations for each expression: a) To factorize 3x - 6y + 15xy, we find the greatest common factor (GCF) of the coefficients. Step 1: Identify the GCF of 3, 6, and 15, which is 3. 3x - 6y + 15xy = 3(x - 2y + 5xy) The factored expression is 3(x - 2y + 5xy). b) To factorize x^3 - xy^2, we find the greatest common factor (GCF) of the terms. Step 1: Identify the GCF, which is x. x^3 - xy^2 = x(x^2 - y^2) Step 2: Recognize that x^2 - y^2 is a difference of squares, which factors as (x-y)(x+y). x(x^2 - y^2) = x(x-y)(x+y) The factored expression is x(x-y)(x+y). c) To factorize 3a^2(2x-y) + 12y - 24x, we first rearrange the last two terms to find a common factor. Step 1: Factor out -12 from the last two terms. 12y - 24x = -12(2x - y) Step 2: Substitute this back into the original expression. 3a^2(2x-y) - 12(2x-y) Step 3: Factor out the common binomial factor (2x-y). (2x-y)(3a^2 - 12) Step 4: Factor out the GCF from (3a^2 - 12), which is 3. (2x-y)3(a^2 - 4) Step 5: Recognize that a^2 - 4 is a difference of squares, which factors as (a-2)(a+2). 3(2x-y)(a-2)(a+2) The factored expression is 3(2x-y)(a-2)(a+2). d) To factorize x^4 - 16x^2 - 36, we treat it as a quadratic in terms of x^2. Let u = x^2. Step 1: Rewrite the expression in terms of u. u^2 - 16u - 36 Step 2: Factor the quadratic. We need two numbers that multiply to -36 and add to -16. These numbers are -18 and 2. (u - 18)(u + 2) Step 3: Substitute x^2 back for u. (x^2 - 18)(x^2 + 2) The factored expression is (x^2 - 18)(x^2 + 2). e) To factorize 3x^2 - 15x + 12, we first find the greatest common factor (GCF). Step 1: Identify the GCF of 3, 15, and 12, which is 3. 3(x^2 - 5x + 4) Step 2: Factor the quadratic expression x^2 - 5x + 4. We need two numbers that multiply to 4 and add to -5. These numbers are -1 and -4. 3(x - 1)(x - 4) The factored expression is 3(x - 1)(x - 4). f) To factorize 100x - x^3, we first find the greatest common factor (GCF). Step 1: Identify the GCF, which is x. x(100 - x^2) Step 2: Recognize that 100 - x^2 is a difference of squares, which factors as (10-x)(10+x). x(10 - x)(10 + x) The factored expression is x(10 - x)(10 + x). Send me the next one 📸