Question

Using factor theorem, factorize the polynomials:

x⁴ – 2x³ – 7x² + 8x + 12

Answer 1

f(x) = x⁴ – 2x³ – 7x² + 8x + 12 

Constant term = 12 

Factors of 12 are ±1, ±2, ±3, ±4, ±6, ±12 

Let x – 1 = 0 or x = 1 

f(1) = (1)⁴ – 2(1)³ – 7(1)² + 8(1) + 12 

= 1 – 2 – 7 + 8 + 12 

= 12 

f(1) ≠ 0 

Let x + 1 = 0 or x = -1 

f(-1) = (-1)⁴ – 2(-1)³ – 7(-1)² + 8(-1) + 12 

= 1 + 2 – 7 – 8 + 12

= 0 

f(-1) = 0 

Let x + 2 = 0 or x = -2 

f(-2) = (-2)⁴ – 2(-2)³ – 7(-2)² + 8(-2) + 12

= 16 + 16 – 28 – 16 + 12

= 0 

f(-2) = 0

Let x – 2 = 0 or x = 2 

f(2) = (2)⁴ – 2(2)³ – 7(2)² + 8(2) + 12

= 16 – 16 – 28 + 16 + 12

= 0 

f(2) = 0 

Let x – 3 = 0 or x = 3 

f(3) = (3)⁴ – 2(3)³ – 7(3)² + 8(3) + 12 

= 0 

f(3) = 0 

Therefore, (x – 1), (x + 2), (x – 2) and (x - 3) are factors of f(x) 

Hence f(x) = (x – 1)(x + 2) (x – 2) (x - 3).