Using Synthetic Division, What is the Quotient of 2x³ + 2x + 12 ÷ (x – 2)?

To find the quotient of the polynomial 2x³ + 2x + 12 divided by (x – 2) using synthetic division, we first set up our synthetic division process.

1. Identify the coefficients of the polynomial: For 2x³ + 0x² + 2x + 12, the coefficients are [2, 0, 2, 12].

2. Since we are dividing by (x – 2), we will use 2 for synthetic division.

3. Write down the coefficients:

  2   0   2   12
2 |

4. Bring down the first coefficient:

  2   0   2   12
2 |   2
   |_______

5. Multiply by 2 and add it to the next coefficient:

  2   0   2   12
2 |   2   4
   |_______
     2   4

6. Repeat the process:

  2   0   2   12
2 |   2   4   12
   |_______
     2   4   24

7. Finally, add the last column:

  2   0   2   12
2 |   2   4   12
   |_______
      2   4   24

The final row gives us the coefficients of the quotient, which is 2x² + 4x + 24. Thus, the quotient of 2x³ + 2x + 12 divided by (x – 2) is:

Quotient: 2x² + 4x + 24

So, using synthetic division, we have found that the quotient is 2x² + 4x + 24.