What is the quotient of the polynomial equation x³ + 3x² + 5x + 3 divided by x + 1?

Determining the Quotient of the Polynomial Division

To find the quotient of the polynomial x³ + 3x² + 5x + 3 when divided by x + 1, we will use polynomial long division.

Step-by-Step Polynomial Long Division

1. Setup: Write x³ + 3x² + 5x + 3 inside the division symbol and x + 1 outside.
2. Divide: Determine how many times x in x + 1 goes into the leading term x³. This is x².
   • Multiply x² by x + 1: x²(x + 1) = x³ + x².
3. Subtract: Subtract (x³ + x²) from (x³ + 3x² + 5x + 3):
   • (x³ + 3x² + 5x + 3) – (x³ + x²) = (3x² – x²) + 5x + 3 = 2x² + 5x + 3
4. Repeat: Continue the process:
   • How many times does x go into 2x²? This is 2x.
   • Multiply: 2x(x + 1) = 2x² + 2x.
   • Subtract: (2x² + 5x + 3) – (2x² + 2x) = (5x – 2x) + 3 = 3x + 3.
5. Continue: How many times does x go into 3x? The answer is 3.
   • Multiply: 3(x + 1) = 3x + 3.
   • Subtract: (3x + 3) – (3x + 3) = 0.

Final Quotient

The division process yields no remainder. Therefore, the quotient of the polynomial division (x³ + 3x² + 5x + 3) ÷ (x + 1) is:

x² + 2x + 3