(2x^3-5x-7)/(x-2)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (2x^3 - 5x - 7) / (x - 2)

This article will explore the process of dividing the polynomial (2x^3 - 5x - 7) by the binomial (x - 2). We will use polynomial long division to accomplish this task.

Understanding Polynomial Long Division

Polynomial long division is similar to long division with numbers. We aim to find a quotient polynomial and a remainder that satisfy the equation:

Dividend = Divisor * Quotient + Remainder

In our case:

• Dividend: 2x^3 - 5x - 7
• Divisor: x - 2

Steps for Polynomial Long Division

1. Set up the division:

        ________
   x - 2 | 2x^3 - 5x - 7 
   

2. Focus on the leading terms: Divide the leading term of the dividend (2x^3) by the leading term of the divisor (x):

        2x^2 ______
   x - 2 | 2x^3 - 5x - 7 
   

3. Multiply the quotient by the divisor: Multiply the quotient term (2x^2) by the entire divisor (x - 2):

        2x^2 ______
   x - 2 | 2x^3 - 5x - 7 
          2x^3 - 4x^2
   

4. Subtract: Subtract the result from the dividend:

        2x^2 ______
   x - 2 | 2x^3 - 5x - 7 
          2x^3 - 4x^2
          -----------
                4x^2 - 5x 
   

5. Bring down the next term: Bring down the next term from the dividend (-7):

        2x^2 ______
   x - 2 | 2x^3 - 5x - 7 
          2x^3 - 4x^2
          -----------
                4x^2 - 5x - 7
   

6. Repeat steps 2-5: Now repeat the process with the new polynomial (4x^2 - 5x - 7):

   • Divide the leading term (4x^2) by the leading term of the divisor (x): 4x
   • Multiply 4x by (x - 2): 4x^2 - 8x
   • Subtract:

       2x^2 + 4x ______
   x - 2 | 2x^3 - 5x - 7 
         2x^3 - 4x^2
         -----------
               4x^2 - 5x - 7
               4x^2 - 8x
               --------
                     3x - 7
   

   • Bring down the next term (-7)

7. Final step: Continue the process until the degree of the remainder is less than the degree of the divisor. In this case, we're left with:

       2x^2 + 4x + 3 ______
   x - 2 | 2x^3 - 5x - 7 
         2x^3 - 4x^2
         -----------
               4x^2 - 5x - 7
               4x^2 - 8x
               --------
                     3x - 7
                     3x - 6
                     ----
                          -1
   

Result

Therefore, the division of (2x^3 - 5x - 7) by (x - 2) results in:

• Quotient: 2x^2 + 4x + 3
• Remainder: -1

We can express this as:

(2x^3 - 5x - 7) / (x - 2) = 2x^2 + 4x + 3 - (1/(x - 2))