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
Step 1: Write the given expression. The expression provided is: ( (x^3 - 3x^2 + 2x + 1)/(x^2 - 4x + 4) ) dx ( (dx)/(dy) ) + ( (x^2 + 4y - y^2)/([3]4y + y^2) ) Step 2: Simplify the rational function in the first term. The denominator of the first term is x^2 - 4x + 4, which can be factored as (x-2)^2. We perform polynomial long division for (x^3 - 3x^2 + 2x + 1)/(x^2 - 4x + 4): c|cc cc 2rx & +1 \\ 2-5 x^2-4x+4 & x^3 & -3x^2 & +2x & +1 \\ 2r-(x^3 & -4x^2 & +4x) \\ 2-4 2r0 & x^2 & -2x & +1 \\ 2r & -(x^2 & -4x & +4) \\ 3-5 2r & 0 & 2x & -3 \\ So, the rational function simplifies to x + 1 + (2x - 3)/(x^2 - 4x + 4), or x + 1 + (2x - 3)/((x-2)^2). Step 3: Substitute the simplified rational function back into the original expression. Substituting the simplified form into the original expression gives: ( x + 1 + (2x - 3)/((x-2)^2) ) dx ( (dx)/(dy) ) + ( (x^2 + 4y - y^2)/([3]4y + y^2) ) The simplified expression is: ( x + 1 + (2x - 3)/((x-2)^2) ) dx ( (dx)/(dy) ) + ( (x^2 + 4y - y^2)/([3]4y + y^2) ) Send me the next one 📸