(2x^3-5x^2+3x+7)/(x-2) Synthetic Division

Jasonbradley

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Jun 16, 2024

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Solving (2x^3 - 5x^2 + 3x + 7) / (x - 2) Using Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear expression of the form (x - a). It offers a more efficient way to perform polynomial division compared to the traditional long division method.

Let's illustrate how to use synthetic division to solve (2x^3 - 5x^2 + 3x + 7) / (x - 2).

Steps:

1. Identify the coefficients of the polynomial: In our example, the coefficients are 2, -5, 3, and 7.

2. Identify the value of 'a' from the divisor: Our divisor is (x - 2), so a = 2.

3. Set up the synthetic division table: Write the value of 'a' to the left of a horizontal line. Place the coefficients of the polynomial to the right of the line.

       2 | 2  -5   3   7
          ------------------
   

4. Bring down the first coefficient: Bring down the first coefficient (2) below the line.

       2 | 2  -5   3   7
          ------------------
            2
   

5. Multiply and add:

   • Multiply the number you just brought down (2) by the value of 'a' (2), resulting in 4.
   • Write the product (4) under the next coefficient (-5).
   • Add the two numbers (-5 + 4 = -1).

       2 | 2  -5   3   7
          ------------------
            2  -1
   

6. Repeat steps 5 and 6 for the remaining coefficients:

   • Multiply -1 by 2, resulting in -2. Write -2 under 3.
   • Add 3 and -2, resulting in 1.
   • Multiply 1 by 2, resulting in 2. Write 2 under 7.
   • Add 7 and 2, resulting in 9.

       2 | 2  -5   3   7
          ------------------
            2  -1   1   9
   

7. Interpret the results:

   • The numbers below the line represent the coefficients of the quotient polynomial.
   • The last number (9) is the remainder.

Therefore, the result of dividing (2x^3 - 5x^2 + 3x + 7) by (x - 2) is:

2x² - x + 1 + 9 / (x - 2)

Conclusion:

Synthetic division offers a concise and efficient method for dividing polynomials by linear expressions. By following the simple steps outlined above, you can easily solve polynomial divisions without the complexity of traditional long division.