To find the quotient when dividing the polynomial 2x4 + 3x3 + 6x2 + 11x + 8 by (x + 2), we can use polynomial long division.
1. **Set up the division:** Write 2x4 + 3x3 + 6x2 + 11x + 8 under the long division symbol and (x + 2) outside.
2. **Divide the first term:** Divide the leading term of the dividend (2x4) by the leading term of the divisor (x). This gives us 2x3.
3. **Multiply and subtract:** Multiply 2x3 by (x + 2) which gives 2x4 + 4x3. Subtract this from the original polynomial:
- 2x4 + 3x3 + 6x2 + 11x + 8
- -(2x4 + 4x3)
- ———————————–
- -x3 + 6x2 + 11x + 8
4. **Repeat the process:** Next, repeat the division process. Divide -x3 by x to get -x2. Multiply and subtract again:
- -x3 + 6x2 + 11x + 8
- – (-x3 – 2x2)
- ———————————–
- 8x2 + 11x + 8
5. **Continue until all terms are processed:** Keep dividing, multiplying, and subtracting. After fully processing, we get:
The final quotient is: 2x3 – x2 + 4 and a remainder which can be represented as:
Remainder = (R)/(x + 2) where R is the value left after division.
This method helps break down complex polynomials into simpler parts, making it easier to understand how they relate to one another through division.