Using factor theorem, factorize each of the following polynomial:
x4-2x3-7x2+8x+12
Let, f (x) = x4-2x3-7x2+8x+12
The constant term in f (x) is equal to +12 and factors of +12 are 
, 
Putting x = - 1 in f (x), we have
f (-1) = (-1)4 – 2 (-1)3 – 7 (-1)2 + 8 (-1) + 12
= 1 + 2 – 7 – 8 + 12
= 0
Therefore, (x + 1) is a factor of f (x).
Similarly, (x + 2), (x – 2) and (x - 3) are the factors of f (x).
Since, f (x) is a polynomial of degree 4. So, it cannot have more than four linear factors.
Therefore, f (x) = k (x + 1) (x + 2) (x - 2) (x - 3)
x4-2x3-7x2+8x+12 = k (x + 1) (x + 2) (x - 2) (x - 3)
Putting x = 0 on both sides, we get
0 - 0 – 0 + 0 + 12 = k (0 + 1) (0 + 2) (0 - 2) (0 - 3)
12 = 12k
k = 1
Putting k = 1 in f (x) = k (x + 1) (x + 2) (x - 2) (x - 3), we get
f (x) = (x + 1) (x + 2) (x - 2) (x - 3)
Hence,
x4-2x3-7x2+8x+12 = (x + 1) (x + 2) (x - 2) (x - 3)