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
here are the factorizations for the given expressions. i) We need to factorize 125x^3 + y^3. This expression is in the form of a^3 + b^3. We use the identity a^3 + b^3 = (a+b)(a^2 - ab + b^2). Step 1: Identify a and b. We can write 125x^3 as (5x)^3. So, a = 5x. And y^3 is already in the form b^3, so b = y. Step 2: Substitute a and b into the identity. 125x^3 + y^3 = (5x + y)((5x)^2 - (5x)(y) + y^2) Step 3: Simplify the terms. = (5x + y)(25x^2 - 5xy + y^2) The factored form is: (5x + y)(25x^2 - 5xy + y^2) ii) We need to factorize 8x^3 - y^3. This expression is in the form of a^3 - b^3. We use the identity a^3 - b^3 = (a-b)(a^2 + ab + b^2). Step 1: Identify a and b. We can write 8x^3 as (2x)^3. So, a = 2x. And y^3 is already in the form b^3, so b = y. Step 2: Substitute a and b into the identity. 8x^3 - y^3 = (2x - y)((2x)^2 + (2x)(y) + y^2) Step 3: Simplify the terms. = (2x - y)(4x^2 + 2xy + y^2) The factored form is: (2x - y)(4x^2 + 2xy + y^2) iii) We need to factorize (2x - 5y)^3 - (2x - y)^3. This expression is in the form of A^3 - B^3. We use the identity A^3 - B^3 = (A-B)(A^2 + AB + B^2). Step 1: Identify A and B. Let A = (2x - 5y) and B = (2x - y). Step 2: Calculate A-B. A-B = (2x - 5y) - (2x - y) = 2x - 5y - 2x + y = -4y Step 3: Calculate A^2, B^2, and AB. A^2 = (2x - 5y)^2 = (2x)^2 - 2(2x)(5y) + (5y)^2 = 4x^2 - 20xy + 25y^2 B^2 = (2x - y)^2 = (2x)^2 - 2(2x)(y) + (y)^2 = 4x^2 - 4xy + y^2 AB = (2x - 5y)(2x - y) = 2x(2x - y) - 5y(2x - y) = 4x^2 - 2xy - 10xy + 5y^2 = 4x^2 - 12xy + 5y^2 Step 4: Calculate A^2 + AB + B^2. A^2 + AB + B^2 = (4x^2 - 20xy + 25y^2) + (4x^2 - 12xy + 5y^2) + (4x^2 - 4xy + y^2) Combine like terms: = (4x^2 + 4x^2 + 4x^2) + (-20xy - 12xy - 4xy) + (25y^2 + 5y^2 + y^2) = 12x^2 - 36xy + 31y^2 Step 5: Substitute the results from Step 2 and Step 4 into the identity A^3 - B^3 = (A-B)(A^2 + AB + B^2). (2x - 5y)^3 - (2x - y)^3 = (-4y)(12x^2 - 36xy + 31y^2) Step 6: Distribute -4y into the trinomial. = -4y · 12x^2 - 4y · (-36xy) - 4y · 31y^2 = -48x^2y + 144xy^2 - 124y^3 The factored form is: -48x^2y + 144xy^2 - 124y^3 Drop the next question!