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
To factorise the expression 3x^3 - 12xy^2 completely, we will find common factors and apply algebraic identities. Step 1: Identify the greatest common factor (GCF). The terms are 3x^3 and -12xy^2. The numerical coefficients are 3 and -12. The GCF of 3 and 12 is 3. The variable parts are x^3 and x. The GCF of x^3 and x is x. There is no common factor for y^2 as it only appears in the second term. So, the GCF of the entire expression is 3x. Step 2: Factor out the GCF. Divide each term by 3x: (3x^3)/(3x) = x^2 (-12xy^2)/(3x) = -4y^2 So, the expression becomes 3x(x^2 - 4y^2). Step 3: Apply the difference of squares identity. The term inside the parentheses, x^2 - 4y^2, is in the form of a^2 - b^2. Here, a = x and b = sqrt(4y^2) = 2y. The difference of squares identity is a^2 - b^2 = (a-b)(a+b). So, x^2 - 4y^2 = (x - 2y)(x + 2y). Step 4: Combine all factors. Substitute the factored form of x^2 - 4y^2 back into the expression from Step 2. 3x(x^2 - 4y^2) = 3x(x - 2y)(x + 2y) The completely factorised expression is 3x(x - 2y)(x + 2y). 3x(x - 2y)(x + 2y) That's 2 down. 3 left today — send the next one.