2026 · Q34
What should be added to (x²−1)/(x²+1) to get (2x³+3x²−1)/((x+2)(x²+1))?
Elementary MathematicsAlgebra
- A(x²−1)/((x+2)(x²+1))
- B(x²+2)/((x+2)(x²+1))
- C(x+1)/(x+2)
- D(x−1)/(x+2)
Explanation
The required addition is (2x³ + 3x² − 1)/[(x + 2)(x² + 1)] − (x² − 1)/(x² + 1). Combining over the common denominator gives [2x³ + 3x² − 1 − (x² − 1)(x + 2)]/[(x + 2)(x² + 1)] = (x³ + x² + x + 1)/[(x + 2)(x² + 1)]. Since x³ + x² + x + 1 = (x² + 1)(x + 1), this simplifies to (x + 1)/(x + 2), option C.