Question

Using factor theorem, factorize each of the following polynomial:

x³ - 10x² - 53x - 42

Answer 1

Let, f (x) = x³ - 10x² - 53x - 42

The factors of the constant term – 42 are \(\pm\) 1, \(\pm\) 2, \(\pm\) 3, \(\pm\) 6, \(\pm\) 7, \(\pm\) 14, \(\pm\) 21 and \(\pm\) 42

Putting x = - 1, we have

f (-1) = (-1)³ – 10 (-1)² – 53 (-1) - 42

= -1 – 10 + 53 - 42

= 0

So, 

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

Let us now divide

f (x) = x³ - 10x² - 53x - 42 by (x + 1) to get the other factors of f (x)

Using long division method, we get

x³ - 10x² - 53x - 42 = (x + 1) (x² – 11x – 42)

x² – 11x - 42 = x² – 14x + 3x - 42

= x (x – 14) + 3 (x – 14)

= (x – 14) (x + 3)

Hence, 

x³ - 10x² - 53x - 42 = (x + 1) (x - 14) (x + 3)