Synthetic Division: Find Quotient & Remainder (9x^2 - 8) / (x - 3)

Synthetic Division: Finding the Quotient and Remainder of (9x^2 - 8) / (x - 3)

Synthetic division is a shortcut method for dividing polynomials. Let's break down how to find the quotient and remainder when dividing 9x^2 - 8 by x - 3.

Steps:

1. Set up the Synthetic Division Table:

   3 | 9 0 -8

   • We write the divisor (x - 3) as a constant, 3. - The coefficients of the dividend (9x^2 - 8) are written in the table, including any missing terms (in this case, the x term with coefficient 0).

2. Bring Down the First Coefficient:

   3 | 9 0 -8 ----- 9

3. Multiply and Write the Result:

   3 | 9 0 -8 ----- 9 27

   • Multiply the divisor (3) by the number in the bottom row (9) and write the result (27) in the next row.

4. Add the Numbers:

   3 | 9 0 -8 ----- 9 27

   • Add the numbers in the second row.

5. Repeat Steps 3 and 4:

   3 | 9 0 -8 ----- 9 27 81

   • Multiply the divisor (3) by the sum in the second row (27) and write the result (81) in the next row. - Add the numbers in the third row.

6. Continue until All Coefficients are Used:

   3 | 9 0 -8 ----- 9 27 81 ----- 9 27 73

7. Interpret the Results:

   • The numbers in the bottom row (9, 27, 73) represent the coefficients of the quotient, starting with the x term. - The last number (73) is the remainder.

Therefore, the quotient is 9x + 27, and the remainder is 73.

This means that:

(9x^2 - 8) / (x - 3) = 9x + 27 + (73 / (x - 3))