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
We need to factorise completely the given algebraic expressions. a) Factorise 12x^2 - 8xy Step 1: Find the greatest common factor (GCF) of the numerical coefficients. The GCF of 12 and 8 is 4. Step 2: Find the greatest common factor of the variable terms. The terms are x^2 and xy. The common variable factor is x. Step 3: Combine the GCFs from Step 1 and Step 2. The overall GCF is 4x. Step 4: Factor out the GCF from each term. 12x^2 - 8xy = 4x(3x) - 4x(2y) 12x^2 - 8xy = 4x(3x - 2y) The final answer is 4x(3x - 2y). b) Factorise x^2 + 8x + 15 Step 1: Identify two numbers that multiply to the constant term (15) and add up to the coefficient of the middle term (8). We are looking for two numbers, let's call them p and q, such that p × q = 15 and p + q = 8. Step 2: List pairs of factors for 15 and check their sums. • 1 × 15 = 15, and 1 + 15 = 16 (not 8) • 3 × 5 = 15, and 3 + 5 = 8 (this is correct) The two numbers are 3 and 5. Step 3: Write the factored form using these numbers. x^2 + 8x + 15 = (x + 3)(x + 5) The final answer is (x + 3)(x + 5). c) Factorise 2x^2 + 7x + 3 Step 1: Multiply the coefficient of x^2 (a) by the constant term (c). a × c = 2 × 3 = 6 Step 2: Find two numbers that multiply to 6 and add up to the coefficient of the middle term (7). We are looking for two numbers, p and q, such that p × q = 6 and p + q = 7. • 1 × 6 = 6, and 1 + 6 = 7 (this is correct) The two numbers are 1 and 6. Step 3: Rewrite the middle term (7x) using these two numbers. 2x^2 + 7x + 3 = 2x^2 + 1x + 6x + 3 Step 4: Factor by grouping the first two terms and the last two terms. (2x^2 + 1x) + (6x + 3) Factor out the GCF from each group: x(2x + 1) + 3(2x + 1) Step 5: Factor out the common binomial factor (2x + 1). (2x + 1)(x + 3) The final answer is (2x + 1)(x + 3). d) Factorise ac + ad + bc + bd Step 1: Group the terms into pairs. (ac + ad) + (bc + bd) Step 2: Factor out the greatest common factor (GCF) from each pair. From the first pair (ac + ad), the GCF is a. a(c + d) From the second pair (bc + bd), the GCF is b. b(c + d) Step 3: Rewrite the expression with the factored pairs. a(c + d) + b(c + d) Step 4: Factor out the common binomial factor (c + d). (c + d)(a + b) The final answer is (a + b)(c + d). Drop the next question.