Question

Using factor theorem, factorize the polynomials:

x⁴ – 7x³ + 9x² + 7x - 10

Answer 1

Let f(x) = x⁴ – 7x³ + 9x² + 7x- 10 

Constant term = -10 

Factors of -10 are ±1, ±2, ±5, ±10 

Let x – 1 = 0 or x = 1 

f(1) = (1)⁴ – 7(1)³ + 9(1)² + 7(1) – 10 

= 1 – 7 + 9 + 7 - 10 

= 0 

f(1) = 0 

Let x + 1 = 0 or x = -1 

f(-1) = (-1)⁴ – 7(-1)³ + 9(-1)² + 7(-1) – 10 

= 1 + 7 + 9 – 7 - 10

= 0 

f(-1) = 0

Let x – 2 = 0 or x = 2 

f(2) = (2)⁴ – 7(2)³ + 9(2)² + 7(2) – 10 

= 16 – 56 + 36 + 14 – 10

= 0 

f(2) = 0 

Let x – 5 = 0 or x = 5 

f(5) = (5)⁴ – 7(5)³ + 9(5)² + 7(5) – 10 

= 625 – 875 + 225 + 35 – 10

= 0 

f(5) = 0 

Therefore, (x – 1), (x + 1), (x – 2) and (x - 5) are factors of f(x)

Hence f(x) = (x – 1) (x – 1) (x – 2) (x - 5).