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: Simplify the numerator. Combine the like terms in the numerator: 2x + 5x + 2 = 7x + 2 The expression becomes: (7x+2)/(x+1) Step 2: Determine if the fraction is proper or improper. The degree of the numerator (7x+2) is 1. The degree of the denominator (x+1) is 1. Since the degree of the numerator is equal to the degree of the denominator, this is an improper rational function. To perform partial fraction decomposition on an improper fraction, we first perform polynomial long division. Step 3: Perform polynomial long division. Divide 7x+2 by x+1: r 7 \\ x+1 ) 7x+2 \\ -(7x+7) \\ -5 This means that (7x+2)/(x+1) can be written as 7 + (-5)/(x+1). Step 4: Write the partial fraction decomposition. The expression is now in the form of a polynomial plus a proper rational function. The proper rational function (-5)/(x+1) is already in its simplest partial fraction form. Therefore, the partial fraction decomposition is: 7 - (5)/(x+1)