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

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.

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

Set up the synthetic division table:
3  1 0 9 _________
Performing the Division

Bring down the leading coefficient: Bring down the 1 from the dividend.
3  1 0 9 _________ 1

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

Add the column: Add the 0 and 3 to get 3.
3  1 0 9 _________ 1 3

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

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).