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

6 min read Jun 16, 2024
(3x^4-2x^3-x-1)/(x^2-2x+1)

Dividing Polynomials: (3x^4 - 2x^3 - x - 1) / (x^2 - 2x + 1)

In this article, we'll explore how to divide the polynomial (3x^4 - 2x^3 - x - 1) by (x^2 - 2x + 1). We'll employ polynomial long division, a method that mirrors the traditional long division you learned in elementary school.

Understanding the Steps

Let's break down the process step-by-step:

  1. Set up the Division: Arrange the polynomials in descending order of their exponents, filling in any missing terms with a coefficient of zero. We'll write the divisor (x^2 - 2x + 1) to the left and the dividend (3x^4 - 2x^3 - x - 1) to the right, separated by a division symbol.

    x^2 - 2x + 1 | 3x^4 - 2x^3 + 0x^2 - x - 1

  2. Focus on the Leading Terms: Divide the leading term of the dividend (3x^4) by the leading term of the divisor (x^2). This gives us 3x^2. Write this term above the division symbol, aligning it with the x^2 term in the dividend.

        3x^2        
x^2 - 2x + 1 | 3x^4 - 2x^3 + 0x^2 - x - 1

  3. Multiply and Subtract: Multiply the divisor (x^2 - 2x + 1) by the term you just wrote (3x^2). This gives us 3x^4 - 6x^3 + 3x^2. Write this result below the dividend, aligning terms with their corresponding exponents. Subtract the entire expression from the dividend.

        3x^2        
x^2 - 2x + 1 | 3x^4 - 2x^3 + 0x^2 - x - 1 
             -(3x^4 - 6x^3 + 3x^2)
             ---------------------
                  4x^3 - 3x^2 - x

  4. Bring Down the Next Term: Bring down the next term of the dividend (-x).

        3x^2        
x^2 - 2x + 1 | 3x^4 - 2x^3 + 0x^2 - x - 1 
             -(3x^4 - 6x^3 + 3x^2)
             ---------------------
                  4x^3 - 3x^2 - x - 1

  5. Repeat Steps 2-4: Now, repeat the process. Divide the leading term of the new dividend (4x^3) by the leading term of the divisor (x^2), getting 4x. Write this term above the division symbol, aligning it with the x term in the dividend. Multiply the divisor by 4x, subtract, and bring down the next term (-1).

        3x^2 + 4x     
x^2 - 2x + 1 | 3x^4 - 2x^3 + 0x^2 - x - 1 
             -(3x^4 - 6x^3 + 3x^2)
             ---------------------
                  4x^3 - 3x^2 - x - 1
                  -(4x^3 - 8x^2 + 4x)
                  ---------------------
                       5x^2 - 5x - 1

  6. Final Step: Repeat the process one more time. Divide the leading term of the new dividend (5x^2) by the leading term of the divisor (x^2), getting 5. Multiply the divisor by 5, subtract, and you're left with a remainder of 9x - 6.

        3x^2 + 4x + 5   
x^2 - 2x + 1 | 3x^4 - 2x^3 + 0x^2 - x - 1 
             -(3x^4 - 6x^3 + 3x^2)
             ---------------------
                  4x^3 - 3x^2 - x - 1
                  -(4x^3 - 8x^2 + 4x)
                  ---------------------
                       5x^2 - 5x - 1
                       -(5x^2 - 10x + 5)
                       ---------------------
                             9x - 6

The Result

Therefore, the result of dividing (3x^4 - 2x^3 - x - 1) by (x^2 - 2x + 1) is:

(3x^4 - 2x^3 - x - 1) / (x^2 - 2x + 1) = 3x^2 + 4x + 5 + (9x - 6) / (x^2 - 2x + 1)

This can be expressed as the quotient 3x^2 + 4x + 5 with a remainder of 9x - 6.

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