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

6 min read Jun 16, 2024
(6x^4-3x^3+5x^2+2x-6)/(3x^2-2)

Polynomial Long Division: (6x^4-3x^3+5x^2+2x-6)/(3x^2-2)

This article will walk through the process of dividing the polynomial 6x^4 - 3x^3 + 5x^2 + 2x - 6 by 3x^2 - 2 using polynomial long division.

Understanding Polynomial Long Division

Polynomial long division is similar to the long division we learned in elementary school, but instead of working with numbers, we work with polynomials. The goal is to find a quotient polynomial and a remainder polynomial that satisfy the equation:

Dividend = Divisor x Quotient + Remainder

In our case:

  • Dividend: 6x^4 - 3x^3 + 5x^2 + 2x - 6
  • Divisor: 3x^2 - 2

The Steps of Polynomial Long Division

  1. Set up the problem: Write the dividend and divisor in a long division format, similar to how you would do with numbers.

         _________
3x^2-2 | 6x^4 - 3x^3 + 5x^2 + 2x - 6

  2. Divide the leading terms: Divide the leading term of the dividend (6x^4) by the leading term of the divisor (3x^2). This gives us 2x^2. Write this quotient above the dividend, aligning it with the x^2 term.

         2x^2 ______
3x^2-2 | 6x^4 - 3x^3 + 5x^2 + 2x - 6

  3. Multiply the quotient by the divisor: Multiply the quotient (2x^2) by the divisor (3x^2 - 2). This gives us 6x^4 - 4x^2. Write this result below the dividend.

         2x^2 ______
3x^2-2 | 6x^4 - 3x^3 + 5x^2 + 2x - 6 
        6x^4 - 4x^2

  4. Subtract: Subtract the result from step 3 from the dividend. This leaves us with -3x^3 + 9x^2 + 2x - 6.

         2x^2 ______
3x^2-2 | 6x^4 - 3x^3 + 5x^2 + 2x - 6 
        6x^4 - 4x^2
        ----------------
              -3x^3 + 9x^2 + 2x - 6

  5. Bring down the next term: Bring down the next term of the dividend (-6) and add it to the result from step 4.

         2x^2 ______
3x^2-2 | 6x^4 - 3x^3 + 5x^2 + 2x - 6 
        6x^4 - 4x^2
        ----------------
              -3x^3 + 9x^2 + 2x - 6

  6. Repeat steps 2-5: Now we repeat steps 2-5 with the new dividend (-3x^3 + 9x^2 + 2x - 6).

    • Divide the leading term (-3x^3) by the leading term of the divisor (3x^2). This gives us -x.
    • Multiply -x by the divisor (3x^2 - 2): -3x^3 + 2x
    • Subtract this result from -3x^3 + 9x^2 + 2x - 6.
    • Bring down the next term (-6).

    The result is:

         2x^2 - x _____
3x^2-2 | 6x^4 - 3x^3 + 5x^2 + 2x - 6 
        6x^4 - 4x^2
        ----------------
              -3x^3 + 9x^2 + 2x - 6
              -3x^3 + 2x 
              ----------
                    9x^2 + 0x - 6

  7. Repeat steps 2-5 again: Repeat the steps with the new dividend (9x^2 + 0x - 6).

    • Divide 9x^2 by 3x^2, which gives us 3.
    • Multiply 3 by the divisor (3x^2 - 2): 9x^2 - 6.
    • Subtract this from 9x^2 + 0x - 6.

    The result is:

         2x^2 - x + 3 _____
3x^2-2 | 6x^4 - 3x^3 + 5x^2 + 2x - 6 
        6x^4 - 4x^2
        ----------------
              -3x^3 + 9x^2 + 2x - 6
              -3x^3 + 2x 
              ----------
                    9x^2 + 0x - 6
                    9x^2 - 6
                    ---------
                        0x + 0

  8. Stop when the degree of the remainder is less than the degree of the divisor: Since the remainder (0) has a degree of 0, which is less than the degree of the divisor (3x^2 - 2), we stop here.

The Result

Therefore, the quotient is 2x^2 - x + 3 and the remainder is 0.

This means that:

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

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