(x^2+9)/(x-3) Synthetic Division

Jasonbradley

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Jun 17, 2024

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Synthetic Division: (x² + 9) / (x - 3)

Synthetic division is a shortcut method for dividing polynomials, particularly when the divisor is in the form (x - a). Let's explore how to perform synthetic division on the expression (x² + 9) / (x - 3).

Setting up the Problem

1. Identify the coefficients of the dividend: In this case, our dividend is x² + 9. Since there's no x term, we need to include a coefficient of 0 for it: 1, 0, 9.

2. Identify the constant term of the divisor: Our divisor is (x - 3), so the constant term is 3.

3. Set up the synthetic division table:

     3 | 1  0  9
       |_________
   

Performing the Division

1. Bring down the leading coefficient: Bring down the 1 from the dividend.

     3 | 1  0  9
       |_________
           1
   

2. Multiply and add: Multiply the number we just brought down (1) by the divisor's constant term (3), and write the result (3) under the next coefficient (0).

     3 | 1  0  9
       |_________
           1  3
   

3. Add the column: Add the 0 and 3 to get 3.

     3 | 1  0  9
       |_________
           1  3
   

4. Repeat steps 2 and 3: Multiply the new number (3) by the divisor's constant term (3), and write the result (9) under the next coefficient (9).

     3 | 1  0  9
       |_________
           1  3  9
   

5. Add the column: Add the 9 and 9 to get 18.

     3 | 1  0  9
       |_________
           1  3  18
   

Interpreting the Results

The numbers in the bottom row of the table represent the coefficients of the quotient and the remainder.

• Quotient: The quotient is x + 3.
• Remainder: The remainder is 18.

Therefore, the result of the division (x² + 9) / (x - 3) is x + 3 + 18/(x - 3).