Question

Using factor theorem, factorize each of the following polynomial:

x⁴ + 10x³ + 35x² + 50x + 24

Answer 1

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)