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 factorisations: 1.1 2x + 4y - 6z Step 1: Find the greatest common factor (GCF) of all terms. The GCF of 2, 4, and 6 is 2. 2x + 4y - 6z = 2(x) + 2(2y) - 2(3z) = 2(x + 2y - 3z) The factorised expression is 2(x + 2y - 3z). 1.2 10p^6q^2 - 4p^3q^2 + 2p^4q^4 Step 1: Find the greatest common factor (GCF) of all terms. The GCF of the coefficients 10, 4, and 2 is 2. The GCF of the p terms (p^6, p^3, p^4) is p^3. The GCF of the q terms (q^2, q^2, q^4) is q^2. So, the overall GCF is 2p^3q^2. 10p^6q^2 - 4p^3q^2 + 2p^4q^4 = 2p^3q^2(5p^3) - 2p^3q^2(2) + 2p^3q^2(p q^2) = 2p^3q^2(5p^3 - 2 + p q^2) The factorised expression is 2p^3q^2(5p^3 - 2 + p q^2). 1.3 (m+n)^2 - 5p(m+n) Step 1: Identify the common binomial factor, which is (m+n). (m+n)^2 - 5p(m+n) = (m+n)(m+n) - 5p(m+n) = (m+n)[(m+n) - 5p] = (m+n)(m+n - 5p) The factorised expression is (m+n)(m+n - 5p). 1.4 4(7c - d) + a(d - 7c) Step 1: Rewrite the term a(d - 7c) to have a common factor with 4(7c - d). Note that (d - 7c) = -(7c - d). 4(7c - d) + a(d - 7c) = 4(7c - d) + a[-(7c - d)] = 4(7c - d) - a(7c - d) Step 2: Factor out the common binomial factor (7c - d). = (7c - d)(4 - a) The factorised expression is (7c - d)(4 - a). 1.5 3ax - 6bx + 6ay - 12by Step 1: Group the terms and factor out the GCF from each pair. (3ax - 6bx) + (6ay - 12by) Factor 3x from the first pair and 6y from the second pair. 3x(a - 2b) + 6y(a - 2b) Step 2: Factor out the common binomial factor (a - 2b). (a - 2b)(3x + 6y) Step 3: Factor out the GCF from the remaining binomial (3x + 6y). The GCF of 3x and 6y is 3. (a - 2b)3(x + 2y) = 3(a - 2b)(x + 2y) The factorised expression is 3(a - 2b)(x + 2y). That's 2 down. 3 left today — send the next one.