To find the quotient of the polynomial x³ + 3x² + 5x + 3 divided by x + 1, we can use polynomial long division.
1. **Set up the division:** Write x³ + 3x² + 5x + 3 under the long division symbol and x + 1 outside.
2. **Divide the first term:** Take the leading term of the numerator x³ and divide it by the leading term of the denominator x, which gives x².
3. **Multiply and subtract:** Multiply x² by x + 1 (which equals x³ + x²) and subtract this from the original polynomial:
(x³ + 3x² + 5x + 3) – (x³ + x²) = 2x² + 5x + 3.
4. **Repeat the process:** Now take the new leading term 2x², divide by x to get 2x. Multiply 2x by x + 1 (which gives 2x² + 2x), and subtract:
(2x² + 5x + 3) – (2x² + 2x) = 3x + 3.
5. **Continue dividing:** Now take the leading term 3x, divide by x to get 3. Multiply 3 by x + 1 (which gives 3x + 3), and subtract:
(3x + 3) – (3x + 3) = 0.
Now, since the remainder is 0, we conclude that the division is exact. Therefore, the quotient of the division is:
x² + 2x + 3.
In summary, the quotient of x³ + 3x² + 5x + 3 divided by x + 1 is x² + 2x + 3.