Determine the remainder when x^3+3x^2-x-2 is divided by (x+3)(x+5)?

Let (x + a) be the quotient and (bx + c) be the remainder when x³ + 3x² - x - 2 is divided by (x + 3)(x + 5).

Then, x³ + 3x² - x - 2 = (x + 3)(x + 5)(x + a) + (bx + c) ...... [1]

Put x = -3 into [1] :
(-3)³ + 3(-3)² - (-3) - 2 = b(-3) + c
-3b + c = 1 ...... [2]

Put x = -5 into [1] :
(-5)³ + 3(-5)² - (-5) - 2 = b(-5) + c
-5b + c = -47 ...... [3]

[2] - [3] :
2b = 48
b = 24

Put b = 24 into [2] :
-3(24) + c = 1
c = 73

Hence, remainder = 24x + 73

= (x³ + 3x² - x - 2)/[(x + 3).(x + 5)]

= (x³ + 3x² - x - 2)/[x² + 5x + 3x + 15]

= (x³ + 3x² - x - 2)/(x² + 8x + 15)


First term: x³/x² = x

x.(x² + 8x + 15) = x³ + 8x² + 15x

Rest:

= (x³ + 3x² - x - 2) - (x³ + 8x² + 15x)

= x³ + 3x² - x - 2 - x³ - 8x² - 15x

= - 5x² - 16x - 2


Second term: - 5x²/x² = - 5

- 5.(x² + 8x + 15) = - 5x² - 40x - 75

Rest:

= (- 5x² - 16x - 2) - (- 5x² - 40x - 75)

= - 5x² - 16x - 2 + 5x² + 40x + 75

= 24x + 73 ← this is the remainder



(x³ + 3x² - x - 2) = [(x - 5).(x + 3).(x + 5)] + (24x + 73)

(x³ + 3x² - x - 2)/(x + 3).(x + 5) = (x - 5) + [(24x + 73)/(x + 3).(x + 5)]

use long division...
Answer: 24x+73

(x+3)(x+5) = x^2+8x+15

.... ..... ..... ..x-5
---- ----- ----- ---- ---- ----------
x^2+8x+15 ) x^3+3x^2-x-2
... ........x^3+8x^2+15x
--------- -------- --------- --------
..... ...... ...-5x^2-16x-2
.... .... .... .-5x^2-40x-75
---------- ----- ------ ----- ------
....... ........ .....24x+73

The remainder is 24x+73