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(6x^3+10x^2+x+8)/(2x^2+1)

4 min read Jun 16, 2024
(6x^3+10x^2+x+8)/(2x^2+1)

Polynomial Long Division: (6x³ + 10x² + x + 8) ÷ (2x² + 1)

This article explores the process of dividing the polynomial 6x³ + 10x² + x + 8 by the polynomial 2x² + 1 using polynomial long division.

Setting Up the Division

  1. Organize the Polynomials: Write the dividend (6x³ + 10x² + x + 8) and the divisor (2x² + 1) in a long division format, ensuring both polynomials are in descending order of their exponents. Notice that the dividend has a missing x term, so we can include a placeholder '0x' for clarity.

    2x² + 1 | 6x³ + 10x² + x + 8

Performing the Division

  1. Divide the Leading Terms: Divide the leading term of the dividend (6x³) by the leading term of the divisor (2x²), which gives 3x. Write this term above the line in the quotient.

        3x
2x² + 1 | 6x³ + 10x² + x + 8

  2. Multiply the Quotient by the Divisor: Multiply the term just written in the quotient (3x) by the entire divisor (2x² + 1), which results in 6x³ + 3x. Write this product below the dividend.

        3x
2x² + 1 | 6x³ + 10x² + x + 8 
         6x³ + 3x

  3. Subtract: Subtract the product (6x³ + 3x) from the corresponding terms of the dividend. Remember that subtracting a term is the same as adding its opposite.

        3x
2x² + 1 | 6x³ + 10x² + x + 8 
         6x³ + 3x 
         -------
               10x² - 2x + 8

  4. Bring Down the Next Term: Bring down the next term of the dividend (8) to the bottom line.

        3x
2x² + 1 | 6x³ + 10x² + x + 8 
         6x³ + 3x 
         -------
               10x² - 2x + 8

  5. Repeat Steps 1-4: Repeat the process with the new polynomial (10x² - 2x + 8). Divide the leading term of this polynomial (10x²) by the leading term of the divisor (2x²), which gives 5. Write 5 in the quotient.

        3x + 5
2x² + 1 | 6x³ + 10x² + x + 8 
         6x³ + 3x 
         -------
               10x² - 2x + 8 
               10x² + 5

  6. Subtract: Subtract the product (10x² + 5) from the corresponding terms.

        3x + 5
2x² + 1 | 6x³ + 10x² + x + 8 
         6x³ + 3x 
         -------
               10x² - 2x + 8 
               10x² + 5
               -------
                     -2x + 3

  7. Final Step: Since the degree of the remaining polynomial (-2x + 3) is less than the degree of the divisor (2x² + 1), we stop here. The remainder is -2x + 3.

The Result

The result of the division can be expressed as follows:

(6x³ + 10x² + x + 8) ÷ (2x² + 1) = 3x + 5 + (-2x + 3)/(2x² + 1)

This means that:

  • Quotient: 3x + 5
  • Remainder: -2x + 3

Therefore, we have successfully divided the polynomial 6x³ + 10x² + x + 8 by the polynomial 2x² + 1 using long division.

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