Divisibility by (x - 2):
To check if the polynomial is exactly divisible by (x - 2), we need to find the remainder when the polynomial is divided by (x - 2) and equate it to zero.

Using Synthetic Division:
- Write down the coefficients of the polynomial: 1, a, -7, 8, b.
- Change the sign of the divisor: -2 becomes +2.
- Bring down the first coefficient: 1.
- Multiply the divisor by the value just brought down and write it beneath the next coefficient: 2a.
- Add the value in the second column to the next coefficient: a + 2a = 3a.
- Repeat the process until all coefficients have been used.
- The last value obtained should be the remainder.

The synthetic division table looks like this:

2 | 1 a -7 8 b
| 2 4a+8 2a²+4a-14 4a²-20a+16
--------------------------------------
| 1 2+a 2a²+4a-6 4a²-12a+16+b

Since the polynomial is exactly divisible by (x - 2), the remainder should be zero. Therefore, the last term in the table should be zero:

4a² - 12a + 16 + b = 0

Simplifying the equation, we get:

4a² - 12a + b = -16 (Equation 1)

Divisibility by (x - 3):
To check if the polynomial is exactly divisible by (x - 3), we follow the same process as above.

The synthetic division table looks like this:

3 | 1 a -7 8 b
| 3 9a+27 24a+81 72a+216
--------------------------------------
| 1 3+a 9a+17 32a+89+b

Since the polynomial is exactly divisible by (x - 3), the remainder should be zero. Therefore, the last term in the table should be zero:

32a + 89 + b = 0

Simplifying the equation, we get:

32a + b = -89 (Equation 2)

Solving the Equations:
We now have two equations: Equation 1 and Equation 2. By solving these equations simultaneously, we can find the values of a and b.

Equation 1: 4a² - 12a + b = -16
Equation 2: 32a + b = -89

Substituting Equation 2 into Equation 1, we get:

4a² - 12a + (32a + b) = -16
4a² + 20a + b = -16

Simplifying the equation further, we get:

4a² + 20a + b + 16 = 0

Using the quadratic formula, we can solve for 'a':

a = (-20 ± √(400 - 16(b + 16))) / 8

Similarly,