Question

Using factor theorem, factorize each of the following polynomial:

x³ - 6x² + 3x + 10

Answer 1

Let, f (x) = x³ - 6x² + 3x + 10

The constant term in f (x) is equal to 10 and factors of 10 are \(\pm\) 1, \(\pm\) 2, \(\pm\) 5 and \(\pm\) 10

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

f (-1) = (-1)³ – 6 (-1)² + 3 (-1) + 10

= -1 – 6 – 3 + 10

= 0

Therefore, 

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

Similarly, (x - 2) and (x - 5) are the factors of f (x).

Since, f (x) is a polynomial of degree 3. So, it cannot have more than three linear factors.

Therefore, 

f (x) = k (x + 1) (x - 2) (x - 5)

x³ - 6x² + 3x + 10 = k (x + 1) (x - 2) (x - 5)

Putting x = 0 on both sides, we get

0 + 0 – 0 + 10 = k (0 + 1) (0 - 2) (0 - 5)

10 = 10k

k = 1

Putting k = 1 in f (x) = k (x + 1) (x - 2) (x - 5), we get

f (x) = (x + 1) (x - 2) (x - 5)

Hence,

x³ - 6x² + 3x + 10 = (x + 1) (x - 2) (x - 5)