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(8x^3-3x+1)/(4x^3+x^2-2x-3) Long Division

5 min read Jun 16, 2024
(8x^3-3x+1)/(4x^3+x^2-2x-3) Long Division

Long Division of Polynomials: (8x^3 - 3x + 1) / (4x^3 + x^2 - 2x - 3)

Long division of polynomials is a method for dividing one polynomial by another polynomial. It is similar to the long division of numbers. In this article, we will explore how to divide the polynomial (8x^3 - 3x + 1) by (4x^3 + x^2 - 2x - 3).

Setting up the Division

  1. Arrange the polynomials in descending order of their exponents. In our case, both polynomials are already arranged.
  2. Write the dividend (8x^3 - 3x + 1) under the division symbol and the divisor (4x^3 + x^2 - 2x - 3) outside the symbol.
             _________
4x^3 + x^2 - 2x - 3 | 8x^3 - 3x + 1

Performing the Division

  1. Divide the leading term of the dividend (8x^3) by the leading term of the divisor (4x^3). This gives us 2. Write this quotient above the dividend.
  2. Multiply the quotient (2) by the entire divisor (4x^3 + x^2 - 2x - 3). This gives us 8x^3 + 2x^2 - 4x - 6. Write this result below the dividend.
  3. Subtract the result from the dividend.
  4. Bring down the next term of the dividend (1).
             2       
4x^3 + x^2 - 2x - 3 | 8x^3 - 3x + 1 
                    -(8x^3 + 2x^2 - 4x - 6)
                    --------------------
                          -2x^2 + x + 7
  1. Repeat steps 1-4 with the new dividend (-2x^2 + x + 7).
  • Divide the leading term of the new dividend (-2x^2) by the leading term of the divisor (4x^3). This gives us -1/2x.
  • Multiply the new quotient (-1/2x) by the divisor. This gives us -2x^2 - 1/2x + x + 3/2.
  • Subtract this result from the current dividend.
  • Bring down the next term (which is 0, since the original dividend didn't have an x^2 term).
             2 - 1/2x      
4x^3 + x^2 - 2x - 3 | 8x^3 - 3x + 1 
                    -(8x^3 + 2x^2 - 4x - 6)
                    --------------------
                          -2x^2 + x + 7 
                          -(-2x^2 - 1/2x + x + 3/2)
                          --------------------------
                                   3/2x + 11/2
  1. Continue this process until the degree of the remaining dividend is less than the degree of the divisor. In this case, the degree of (3/2x + 11/2) is 1, which is less than the degree of (4x^3 + x^2 - 2x - 3). Therefore, we stop here.

The Result

The result of the long division is:

(8x^3 - 3x + 1) / (4x^3 + x^2 - 2x - 3) = 2 - 1/2x + (3/2x + 11/2) / (4x^3 + x^2 - 2x - 3)

This can also be written as:

(8x^3 - 3x + 1) = (4x^3 + x^2 - 2x - 3)(2 - 1/2x) + (3/2x + 11/2)

This indicates that the original polynomial (8x^3 - 3x + 1) can be expressed as the product of the divisor (4x^3 + x^2 - 2x - 3) and the quotient (2 - 1/2x) plus the remainder (3/2x + 11/2).

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