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

Jasonbradley

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Jun 16, 2024

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Polynomial Long Division: (3x - 4x^3 + 6x^4 + 1) / (x + 3)

This article will guide you through the process of polynomial long division with the specific example of (3x - 4x^3 + 6x^4 + 1) / (x + 3).

Understanding Polynomial Long Division

Polynomial long division is a method used to divide polynomials. It is similar to the long division process you may have learned in elementary school for numbers.

Steps:

1. Set up the division: Arrange the terms of the dividend (the polynomial being divided) and the divisor (the polynomial doing the dividing) in descending order of exponents. If any terms are missing, use a placeholder with a coefficient of zero.

        6x^4 - 4x^3 + 3x + 1 
   x + 3 |____________________
   

2. Divide the leading terms: Divide the leading term of the dividend (6x^4) by the leading term of the divisor (x). This gives us 6x^3. Write this result above the dividend.

        6x^4 - 4x^3 + 3x + 1 
   x + 3 | 6x^3____________________
   

3. Multiply and subtract: Multiply the divisor (x + 3) by the term we just wrote (6x^3). Write the result below the dividend and subtract.

        6x^4 - 4x^3 + 3x + 1 
   x + 3 | 6x^3____________________
              6x^4 + 18x^3 
              -------------
                    -22x^3 + 3x 
   

4. Bring down the next term: Bring down the next term of the dividend (3x).

        6x^4 - 4x^3 + 3x + 1 
   x + 3 | 6x^3____________________
              6x^4 + 18x^3 
              -------------
                    -22x^3 + 3x + 1 
   

5. Repeat steps 2-4: Divide the new leading term (-22x^3) by the leading term of the divisor (x) to get -22x^2. Write this above the dividend. Multiply (-22x^2) by the divisor (x + 3) and subtract the result.

        6x^4 - 4x^3 + 3x + 1 
   x + 3 | 6x^3 - 22x^2 ____________________
              6x^4 + 18x^3 
              -------------
                    -22x^3 + 3x + 1 
                    -22x^3 - 66x^2
                    --------------
                           69x^2 + 3x
   

6. Continue until the degree of the remainder is less than the degree of the divisor: Bring down the next term (1) and continue the process.

        6x^4 - 4x^3 + 3x + 1 
   x + 3 | 6x^3 - 22x^2 + 69x - 206 ____________________
              6x^4 + 18x^3 
              -------------
                    -22x^3 + 3x + 1 
                    -22x^3 - 66x^2
                    --------------
                           69x^2 + 3x + 1 
                           69x^2 + 207x
                           -------------
                                    -204x + 1 
                                    -204x - 612
                                    -------------
                                            613 
   

Result

The result of the long division is:

6x^3 - 22x^2 + 69x - 206 + 613/(x + 3)

This means:

(3x - 4x^3 + 6x^4 + 1) / (x + 3) = 6x^3 - 22x^2 + 69x - 206 + 613/(x + 3)

The final term, 613/(x + 3), represents the remainder.