Question

Using factor theorem, factorize each of the following polynomial:

x³ - 3x² - 9x - 5

Answer 1

Let, f (x) = x³ - 3x² - 9x - 5

The factors of the constant term - 5 are \(\pm\) 1, \(\pm\) 5

Putting x = -1, we have

f (-1) = (-1)³ – 3 (-1)² – 9 (-1) - 5

= -1 – 3 + 9 - 5

= 0

So, 

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

Let us now divide

f (x) = x³ - 3x² - 9x - 5 by (x + 1) to get the other factors of f (x)

Using long division method, we get

x³ - 3x² - 9x - 5 = (x + 1) (x² - 4x 5)

x² - 4x - 5 = x² - 5x + x - 5

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

= (x + 1) (x - 5)

Hence, 

x³ + 13x² + 32x + 20 = (x + 1) (x + 1) (x - 5)

= (x + 1)2 (x – 5)