Q. 63.8( 4 Votes )

Using factor theorem, factorize each of the following polynomial:

x4+10x3+35x2+50x +24

Answer :

Let, f (x) = x4+10x3+35x2+50x +24

The constant term in f (x) is equal to +24 and factors of +24 are ,


Putting x = - 1 in f (x), we have


f (-1) = (-1)4 + 10 (-1)3 + 35 (-1)2 + 50 (-1) + 24


= 1 – 10 + 35 – 50 + 24


= 0


Therefore, (x + 1) is a factor of f (x).


Similarly, (x + 2), (x + 3) and (x + 4) 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 + 3) (x + 4)


x4+10x3+35x2+50x +24 = k (x + 1) (x + 2) (x + 3) (x + 4)


Putting x = 0 on both sides, we get


0 + 0 + 0 + 0 + 24 = k (0 + 1) (0 + 2) (0 + 3) (0 + 4)


24 = 24k


k = 1


Putting k = 1 in f (x) = k (x + 1) (x + 2) (x + 3) (x + 4), we get


f (x) = (x + 1) (x + 2) (x + 3) (x + 4)


Hence,


x4+10x3+35x2+50x +24 = (x + 1) (x + 2) (x + 3) (x + 4)


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