If both x+1 and x-1 are factors of ax3+x2-2x+b, find the value of a and b.
Let, f (X) = ax3+x2-2x+b be the given polynomial.
Given (x + 1) and (x – 1) are factors of f (x).
From factor theorem,
If (x + 1) and (x – 1) are factors of f (x) then f (-1) = 0 and f (1) = 0 respectively.
f (-1) = 0
a (-1)3 + (-1)2 – 2 (-1) + b = 0
-a + 1 + 2 + b = 0
-a + 3 + b = 0
b – a + 3 = 0 (i)
f (1) = 0
a (1)3 + (1)2 – 2 (1) + b = 0
a + 1 – 2 + b = 0
a + b – 1 = 0
b + a – 1 = 0 (ii)
Adding (i) and (ii), we get
b – a + 3 + b + a – 1 = 0
2b + 2 = 0
2b = - 2
b = -1
Putting value of b in (i), we get
-1 - a + 3 = 0
-a + 2 = 0
a = 2
Hence, the value of a = 2 and b = -1.