Answer:
a = - 2, b = - 1
Step-by-step explanation:
If a polynomial f(x) is divisible by (x + h) then f(- h) = 0
P(x) is divisible by (x + 1), thus P(- 1) = 0 , that is
P(- 1) = (- 1)³ + a(- 1)² - b + 2 = 0 , that is
- 1 + a - b + 2 = 0
a - b + 1 = 0 ( subtract 1 from both sides )
a - b = - 1 → (1)
P(x) is divisible by (x - 2) thus P(2) = 0 , that is
P(2) = 2³ + a(2)² + 2b + 2 = 0 , that is
8 + 4a + 2b + 2 = 0
4a + 2b + 10 = 0 ( subtract 10 from both sides )
4a + 2b = - 10 → (2)
Solve (1) and (2) simultaneously
Multiply (1) by 2
2a - 2b = - 2 → (3)
Add (2) and (3) term by term eliminating the term in b
6a = - 12 ( divide both sides by 6 )
a = - 2
Substitute a = - 2 into (1) and solve for b
- 2 - b = - 1 ( add 2 to both sides )
- b = 1 ( multiply both sides by - 1 )
b = - 1
