To find the quotient of the polynomial x³ + 3x² + 5x + 3 by x + 1, we can use polynomial long division.
1. Start by dividing the first term of the dividend (x³) by the first term of the divisor (x). This gives us x².
2. Multiply x² by the entire divisor (x + 1): x² * (x + 1) = x³ + x².
3. Subtract this result from the original polynomial: (x³ + 3x² + 5x + 3) – (x³ + x²) = 2x² + 5x + 3.
4. Repeat the process with the new polynomial 2x² + 5x + 3. Divide 2x² by x to get 2x.
5. Multiply 2x by the divisor: 2x * (x + 1) = 2x² + 2x.
6. Subtract again: (2x² + 5x + 3) – (2x² + 2x) = 3x + 3.
7. Now divide 3x by x to get 3.
8. Multiply 3 by the divisor: 3 * (x + 1) = 3x + 3.
9. Finally, subtract: (3x + 3) – (3x + 3) = 0.
So, the final result of the division is:
x² + 2x + 3
There is no remainder, which means x + 1 is a factor of the polynomial x³ + 3x² + 5x + 3.