Answer :

Let, f (x) = *x*^{4}+10*x*^{3}+35*x*^{2}+50*x* +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)

*x*^{4}+10*x*^{3}+35*x*^{2}+50*x* +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,

*x*^{4}+10*x*^{3}+35*x*^{2}+50*x* +24 = (x + 1) (x + 2) (x + 3) (x + 4)

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