Answer:
a = - 5 and b = 30
Step-by-step explanation:
Using the Remainder theorem
If a polynomial p(x) is divided by (x - h) then
p(h) = Remainder
If (x - h) is a factor of p(x) then p(h) = 0
Given (x - 2) is a factor the p(2) = 0
Given (x + 1) leaves a remainder of - 12 then p(- 1) = - 12
Hence
p(2) = a(2)³ - 11(2) + b(2) + 2 = 0 , that is
8a - 22 + 2b + 2 = 0
8a + 2b - 20 = 0 ( add 20 to both sides )
8a + 2b = 20 → (1)
and
p(- 1) = a(- 1)³ - 11(- 1) + b(- 1) + 2 = - 12, that is
- a + 11 - b + 2 = - 12
- a - b + 13 = - 12 ( subtract 13 from both sides )
- a - b = - 25 ( multiply through by - 1 )
a + b = 25 ( subtract b from both sides )
a = 25 - b → (2)
Substitute a = 25 - b into (1)
8(25 - b) + 2b = 20
200 - 8b + 2b = 20
200 - 6b = 20 ( subtract 200 from both sides )
- 6b = - 180 ( divide both sides by - 6 )
b = 30
Substitute b = 30 into (2)
a = 25 - 30 = - 5
