%2F(x%5E2%2B2x-1).jpeg "(3x^4+2x^3-3x^2+12x-6)/(x^2+2x-1)")
Polynomial Long Division: (3x^4+2x^3-3x^2+12x-6)/(x^2+2x-1)
This article will demonstrate how to perform polynomial long division to find the quotient and remainder of the following expression:
(3x^4+2x^3-3x^2+12x-6)/(x^2+2x-1)
Setting up the Division
First, we set up the division problem similar to traditional long division:
_________
x^2+2x-1 | 3x^4 + 2x^3 - 3x^2 + 12x - 6
Steps for Polynomial Long Division
- Divide the leading terms: Divide the leading term of the dividend (3x^4) by the leading term of the divisor (x^2) to get 3x^2. Write this above the dividend.
3x^2
x^2+2x-1 | 3x^4 + 2x^3 - 3x^2 + 12x - 6
2. **Multiply the quotient by the divisor:** Multiply the quotient (3x^2) by the entire divisor (x^2+2x-1) and write the result below the dividend.
3x^2
x^2+2x-1 | 3x^4 + 2x^3 - 3x^2 + 12x - 6 3x^4 + 6x^3 - 3x^2
3. **Subtract:** Subtract the result from the dividend.
3x^2
x^2+2x-1 | 3x^4 + 2x^3 - 3x^2 + 12x - 6 3x^4 + 6x^3 - 3x^2 ------------------ -4x^3 + 12x - 6
4. **Bring down the next term:** Bring down the next term of the dividend (+12x).
3x^2
x^2+2x-1 | 3x^4 + 2x^3 - 3x^2 + 12x - 6 3x^4 + 6x^3 - 3x^2 ------------------ -4x^3 + 12x - 6
5. **Repeat steps 1-4:** Repeat the process, dividing the leading term of the new dividend (-4x^3) by the leading term of the divisor (x^2) to get -4x. Write this term in the quotient.
3x^2 - 4x
x^2+2x-1 | 3x^4 + 2x^3 - 3x^2 + 12x - 6 3x^4 + 6x^3 - 3x^2 ------------------ -4x^3 + 12x - 6 -4x^3 - 8x^2 + 4x
6. **Subtract and bring down:** Subtract and bring down the next term (-6) to continue the process.
3x^2 - 4x
x^2+2x-1 | 3x^4 + 2x^3 - 3x^2 + 12x - 6 3x^4 + 6x^3 - 3x^2 ------------------ -4x^3 + 12x - 6 -4x^3 - 8x^2 + 4x ------------------- 8x^2 + 8x - 6
7. **Repeat steps 1-4:** Divide the leading term of the new dividend (8x^2) by the leading term of the divisor (x^2) to get 8.
3x^2 - 4x + 8
x^2+2x-1 | 3x^4 + 2x^3 - 3x^2 + 12x - 6 3x^4 + 6x^3 - 3x^2 ------------------ -4x^3 + 12x - 6 -4x^3 - 8x^2 + 4x ------------------- 8x^2 + 8x - 6 8x^2 + 16x - 8
8. **Subtract:** Subtract the result.
3x^2 - 4x + 8
x^2+2x-1 | 3x^4 + 2x^3 - 3x^2 + 12x - 6 3x^4 + 6x^3 - 3x^2 ------------------ -4x^3 + 12x - 6 -4x^3 - 8x^2 + 4x ------------------- 8x^2 + 8x - 6 8x^2 + 16x - 8 ----------------- -8x + 2
9. **The remainder is the final result:** The degree of the remainder (-8x+2) is less than the degree of the divisor (x^2+2x-1), so we stop here.
### Final Result
Therefore, the quotient of the division is **3x^2 - 4x + 8** and the remainder is **-8x + 2**.
The full result can be written as:
**(3x^4+2x^3-3x^2+12x-6)/(x^2+2x-1) = 3x^2 - 4x + 8 + (-8x+2)/(x^2+2x-1)**