To find the quotient of dividing the polynomial x^3 + 5x^2 + 3x + 8 by x + 3, we can use polynomial long division.
1. **Set up the long division**: Write x^3 + 5x^2 + 3x + 8 under the long division bar and x + 3 outside.
2. **Divide the first term**: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x), which gives us x^2.
3. **Multiply and subtract**: Multiply x^2 by x + 3 to get x^3 + 3x^2. Subtract this from the original polynomial:
x^3 + 5x^2 + 3x + 8 – (x^3 + 3x^2) = 2x^2 + 3x + 8.
4. **Repeat the process**: Now divide the leading term of the new polynomial (2x^2) by the leading term of the divisor (x), which gives 2x. Multiply 2x by x + 3 to get 2x^2 + 6x. Subtract:
2x^2 + 3x + 8 – (2x^2 + 6x) = -3x + 8.
5. **Continue**: Now divide -3x by x to get -3. Multiply -3 by x + 3 to get -3x – 9. Subtract:
-3x + 8 – (-3x – 9) = 17.
6. **Complete the division**: Now, we cannot divide 17 by x + 3 as it’s a constant and the divisor is a linear polynomial. Thus, 17 is the remainder.
So, we can summarize the result:
Quotient: x^2 + 2x – 3
Remainder: 17
In conclusion, when x^3 + 5x^2 + 3x + 8 is divided by x + 3, the quotient is x^2 + 2x – 3, with a remainder of 17.