Find the values of a and b so that the polynomial is

exactly divisible by (x+2) as well as (x+3).

Let f(x) = x⁴ + ax³ – 7x² - 8x + b

Now,

(x + 2) = 0

x = -2

And,

(x + 3) = 0

x = -3

Now,

By using factor theorem,

(x + 2) and (x + 3) will be the factors of f(x) if f(-2) = 0 and f(-3) = 0

Hence,

f(-2) = (-2)⁴ + a(-2)³- 7 (-2)² -8 (-2) + b
0 = 16 – 8a - 28 + 16 + b

8a - b = 4 (i)

And,

f(-3) = (-3)⁴ + a (-3)³ – 10 (-3)² – 8 (-3) + b

0 = 81 – 27a – 63 + 24 + b

27a – b = 42 (ii)

Now, subtracting (i) from (ii)

19a = 38

a = 2

Using the value of a in (i), we get

8 (2) – b = 4

16 – b = 4

b = 12

Therefore,

a = 2 and b = 12