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

5 min read Jun 17, 2024
(x^3+3x^2-6x+2)/(x-1)

Polynomial Long Division: (x^3+3x^2-6x+2)/(x-1)

This article explores the process of dividing the polynomial x^3 + 3x^2 - 6x + 2 by x - 1 using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is a method for dividing polynomials, similar to the long division method used for numbers. It helps us to rewrite a polynomial as a product of two polynomials (quotient and divisor) plus a remainder.

Steps for Polynomial Long Division

Let's break down the division of (x^3 + 3x^2 - 6x + 2) by (x - 1):

  1. Set up the division:

         ________
    x - 1 | x^3 + 3x^2 - 6x + 2
    
  2. Divide the leading terms:

    • The leading term of the dividend (x^3) is divided by the leading term of the divisor (x).
    • The result is x^2, which is placed above the x^2 term in the quotient.
         x^2 ______
    x - 1 | x^3 + 3x^2 - 6x + 2
    
  3. Multiply the divisor by the quotient term:

    • Multiply (x - 1) by x^2, which gives x^3 - x^2.
         x^2 ______
    x - 1 | x^3 + 3x^2 - 6x + 2
            x^3 - x^2
    
  4. Subtract:

    • Subtract the product obtained in the previous step from the dividend.
    • Change the signs of the terms in the product and add.
         x^2 ______
    x - 1 | x^3 + 3x^2 - 6x + 2
            x^3 - x^2
            -------
                  4x^2 
    
  5. Bring down the next term:

    • Bring down the next term of the dividend (-6x).
         x^2 ______
    x - 1 | x^3 + 3x^2 - 6x + 2
            x^3 - x^2
            -------
                  4x^2 - 6x 
    
  6. Repeat steps 2-5:

    • Divide the new leading term (4x^2) by the leading term of the divisor (x), which gives 4x.
    • Multiply (x - 1) by 4x to get 4x^2 - 4x.
    • Subtract the product, changing signs and adding.
    • Bring down the next term (+2).
         x^2 + 4x ______
    x - 1 | x^3 + 3x^2 - 6x + 2
            x^3 - x^2
            -------
                  4x^2 - 6x 
                  4x^2 - 4x
                  -------
                        -2x + 2
    
  7. Final step:

    • Divide -2x by x, which gives -2.
    • Multiply (x - 1) by -2 to get -2x + 2.
    • Subtract, changing signs and adding.
         x^2 + 4x - 2 _____
    x - 1 | x^3 + 3x^2 - 6x + 2
            x^3 - x^2
            -------
                  4x^2 - 6x 
                  4x^2 - 4x
                  -------
                        -2x + 2
                        -2x + 2
                        -------
                             0 
    

Result

Therefore, the division of (x^3 + 3x^2 - 6x + 2) by (x - 1) gives us:

Quotient: x^2 + 4x - 2 Remainder: 0

This can be expressed as:

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

Conclusion

Polynomial long division allows us to systematically divide polynomials and express the result as a quotient and a remainder. This method is crucial for simplifying expressions, factoring polynomials, and solving algebraic equations.

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