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(5x^4+2x^3-9x+12)/(x^2-3x+4)

5 min read Jun 16, 2024
(5x^4+2x^3-9x+12)/(x^2-3x+4)

Polynomial Long Division: (5x^4+2x^3-9x+12)/(x^2-3x+4)

This article will guide you through the process of performing polynomial long division on the expression (5x^4+2x^3-9x+12)/(x^2-3x+4).

Understanding Polynomial Long Division

Polynomial long division is a method used to divide polynomials, much like long division is used to divide numbers. The goal is to find the quotient and remainder of the division.

Steps for Polynomial Long Division

  1. Set up the division: Arrange the dividend (5x^4+2x^3-9x+12) and the divisor (x^2-3x+4) in the format of long division.

         _________
x^2-3x+4 | 5x^4 + 2x^3 - 9x + 12

  2. Divide the leading terms: Divide the leading term of the dividend (5x^4) by the leading term of the divisor (x^2). This gives us 5x^2. Write this above the dividend.

         5x^2     
x^2-3x+4 | 5x^4 + 2x^3 - 9x + 12

  3. Multiply the quotient by the divisor: Multiply the quotient (5x^2) by the entire divisor (x^2-3x+4). This gives us 5x^4 - 15x^3 + 20x^2.

         5x^2     
x^2-3x+4 | 5x^4 + 2x^3 - 9x + 12
           5x^4 - 15x^3 + 20x^2

  4. Subtract: Subtract the result from the dividend.

         5x^2     
x^2-3x+4 | 5x^4 + 2x^3 - 9x + 12
           5x^4 - 15x^3 + 20x^2 
           ----------------------
                   17x^3 - 20x^2 - 9x

  5. Bring down the next term: Bring down the next term of the dividend (-9x).

         5x^2     
x^2-3x+4 | 5x^4 + 2x^3 - 9x + 12
           5x^4 - 15x^3 + 20x^2 
           ----------------------
                   17x^3 - 20x^2 - 9x

  6. Repeat steps 2-5: Repeat the process, dividing the new leading term (17x^3) by the leading term of the divisor (x^2) and continuing to bring down terms until the degree of the remainder is less than the degree of the divisor.

         5x^2 + 17x 
x^2-3x+4 | 5x^4 + 2x^3 - 9x + 12
           5x^4 - 15x^3 + 20x^2 
           ----------------------
                   17x^3 - 20x^2 - 9x
                   17x^3 - 51x^2 + 68x 
                   ----------------------
                           31x^2 - 77x + 12

         5x^2 + 17x + 31
x^2-3x+4 | 5x^4 + 2x^3 - 9x + 12
           5x^4 - 15x^3 + 20x^2 
           ----------------------
                   17x^3 - 20x^2 - 9x
                   17x^3 - 51x^2 + 68x 
                   ----------------------
                           31x^2 - 77x + 12
                           31x^2 - 93x + 124
                           ----------------------
                                  16x - 112

  7. Final Result: The quotient is 5x^2 + 17x + 31 and the remainder is 16x - 112. Therefore, we can write the result as:

    (5x^4 + 2x^3 - 9x + 12) / (x^2 - 3x + 4) = 5x^2 + 17x + 31 + (16x - 112) / (x^2 - 3x + 4)

Conclusion

This example demonstrates how to perform polynomial long division. Understanding this method is crucial for simplifying complex polynomial expressions and solving various algebraic problems.

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