Question

Using factor theorem, factorize the polynomials:

x³ + 2x² – x – 2

Answer 1

Let f(x) = x³ + 2x² – x – 2 

Constant term = -2 

Factors of -2 are ±1, ±2 

Let x – 1 = 0 

⇒ x = 1 

Put the value of x in f(x) 

f(1) = (1)³ + 2(1)² – 1 – 2 

= 1 + 2 – 1 – 2 

= 0

So, (x – 1) is factor of f(x)

Let x + 1 = 0 

⇒ x = -1 

Put the value of x in f(x) 

f(-1) = (-1)³ + 2(-1)² – 1 – 2 

= -1 + 2 + 1 – 2 

= 0 

(x + 1) is a factor of f(x) 

Let x + 2 = 0 

⇒ x = -2 

Put the value of x in f(x) 

f(-2) = (-2)³ + 2(-2)² – (-2) – 2 

= -8 + 8 + 2 – 2 

= 0

(x + 2) is a factor of f(x) 

Let x – 2 = 0

⇒ x = 2 

Put the value of x in f(x) 

f(2) = (2)³ + 2(2)² – 2 – 2 

= 8 + 8 – 2 – 2

= 12 ≠ 0 

(x – 2) is not a factor of f(x)

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