Answer:
(x - 1)(3x + 2)
Explanation:
By the Remainder theorem
Given expression is divisible by x - 1 then f(1) = 0
when divided by (x + 1) then f(- 1) = 2
when divided by x - 2 then f(2) = 8
Substitute x = 1, x = - 1, x = 2 into the expression , that is
a + b + c = 0 → (1)
a - b + c = 2 → (2)
4a + 2b + c = 8 → (3)
Subtract (1) from (2) to eliminate a and c
- 2b = 2 ( divide both sides by - 2 )
b = - 1
Subtract (1) from (3) to eliminate c
3a + b = 8 ← substitute b = - 1
3a - 1 = 8 ( add 1 to both sides )
3a = 9 ( divide both sides by 3 )
a = 3
Substitute a = 3, b = - 1 into (1) and evaluate for c
3 - 1 + c = 0
2 + c = 0 ( subtract 2 from both sides )
c = - 2
Then a = 3, b = - 1, c = - 2
Thus
ax² + bx + c
= 3x² - x - 2
= (x - 1)(3x + 2) ← in factored form