find the values of a and b so that the polynomial (x4 + ax2 - 7x2 + 8x + b) is exactly divisible by (x + 2) as well as (x + 3). how to solve this ??

Let f(x) = x⁴ + ax³ - 7x² + 8x + b

It is given that f(x) is exactly divisible by (x+2). So the remainder is f(-2) which is equal to 0.

Also, it is given that f(x) is exactly divisible by (x+3). So the remainder is f(-3) which is equal to 0.

Therefore, we have:

f(-2) = 16 - 8a - 28 - 16 + b = 0 or  -8a + b - 28 = 0   ... (1)

f(-3) = 81 - 27a - 63 - 24 + b = 0 or  -27a + b - 6 = 0 ... (2)

Subtracting (1) from (2),

-19a + 22 = 0

a = 22/19

 

Now, substitute this value of a in any of the equations (1) or (2) to get the value of b.