Let, f (x) = x4 + 10x3 + 35x2 + 50x + 24
The constant term in f (x) is equal to +24 and factors of +24 are +1, +2, +3, +4, +6, +8, +12 and +18
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)