Answer :

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)


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