Using Synthetic Division, What is the Quotient of 2x³ + 3x + 22 Divided by x – 2?

To find the quotient of the polynomial 2x³ + 3x + 22 divided by x – 2 using synthetic division, we first identify the coefficients of the polynomial:

• 2 (for x³)
• 0 (for x², which is missing)
• 3 (for x)
• 22 (constant)

We set up synthetic division by using the root of the divisor x – 2, which is 2. Next, we write the coefficients:

2  0  3  22
   |  2
---------------------

Now, we proceed with synthetic division:

1. Bring down the 2.
2. Multiply 2 (the number we brought down) by 2 (the divisor root) to get 4 and add it to 0 (the next coefficient) to get 4.
3. Multiply 4 by 2 to get 8 and add it to 3 to get 11.
4. Multiply 11 by 2 to get 22 and add it to 22 (the last coefficient) to get 44.

The results of these calculations yield the coefficients of the quotient polynomial and the remainder:

2  4  11
R:  44

This means the quotient polynomial is 2x² + 4x + 11 and the remainder is 44.

Thus, the final answer can be expressed as:

2x² + 4x + 11 with a remainder of 44.