Question:
If both $(x+1)$ and $(x-1)$ are the factors of $a x^{3}+x^{2}-2 x+b$, Find the values of $a$ and $b$
Solution:
Here, $f(x)=a x^{3}+x^{2}-2 x+b$
(x + 1) and (x - 1) are the factors
From factor theorem, if x = 1, -1 are the factors of f(x) then f(1) = 0 and f(-1) = 0
Let, x - 1= 0
⟹ x = -1
Substitute x value in f(x)
$f(1)=a(1)^{3}+(1)^{2}-2(1)+b$
= a + 1 - 2 + b
= a + b - 1 ..... 1
Let, x + 1 = 0
⟹ x = -1
Substitute x value in f(x)
$f(-1)=a(-1)^{3}+(-1)^{2}-2(-1)+b$
= -a + 1 + 2 + b
= -a + b + 3 ..... 2
Solve equations 1 and 2
a + b = 1
-a + b = -3
2b = -2
⟹ b = -1
substitute b value in eq 1
⟹ a - 1 = 1
⟹ a = 1 + 1
⟹ a = 2
The values are a= 2 and b = -1