(3x^4+2x^3-3x^2+12x-6)/(x^2+2x-1)

6 min read Jun 16, 2024
(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

  1. 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)**

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