Question
The polynomial p(x) = x4 - 2x3 + 3x2 - ax + 3a - 7. When divided by (x + 1) leaves the remainder 19. Find the value of a. Also find the remainder, when p(x) is divided by x + 2.
Solution
p(x) = x4 - 2x3 + 3x2 - ax + 3a - 7
By remainder theorem.
p(- 1) = 19 | x + 1 = 0 
x = -1
(- 1)4 - 2 (- 1)3 + 3 (- 1)2 - a(- 1) + 3a - 7 = 19
4a = 20
Also, the remainder when p(x) is divided by x + 2
= p(-2)
= (-2)4 - 2 (-2)3 + 3(-2)2
- a(-2) + 3a - 7
= 16 + 16 + 12 + 2a + 3a - 7
= 5a + 37
= 5(5) + 37
= 25 + 37
= 62
