Understanding the Problem
To make the polynomial (4x^4 - 2x^3 - 6x^2 + 2x + 6) divisible by (2x^2 + x - 1), we need to find a polynomial R(x) such that:
(4x^4 - 2x^3 - 6x^2 + 2x + 6) - R(x) is divisible by (2x^2 + x - 1).
Long Division Method
1. Divide the given polynomial by (2x^2 + x - 1).
2. Calculate the quotient and the remainder from the division.
3. Identify the remainder, which will help us determine what needs to be subtracted.
Finding Remainder
- Perform polynomial long division:
- Divide the leading term of the dividend by the leading term of the divisor.
- Multiply the entire divisor by this result and subtract from the dividend.
- Repeat the process until the degree of the remainder is less than the degree of the divisor.
Conclusion
- After completing the division, you will obtain a remainder, say R.
- To make the polynomial divisible, you must subtract this remainder from the original polynomial.
Final Calculation
- The value of R will be the polynomial that must be subtracted to achieve exact divisibility.
- Thus, the solution is:
(b) R(x) = remainder obtained from the division.
In summary, conduct polynomial long division, find the remainder, and subtract it from the original polynomial for exact divisibility with (2x^2 + x - 1).