Question

Using factor theorem, factorize the polynomials:

2x⁴ – 7x³ – 13x² + 63x – 45

Answer 1

Let f(x) = 2x⁴ – 7x³ – 13x² + 63x – 45 

Constant term = -45 

Factors of -45 are ±1, ±3, ±5, ±9, ±15, ±45

Here coefficient of x⁴ is 2. 

So possible rational roots of f(x) are ±1, ±3, ±5, ±9, ±15, ±45, ±1/2,±3/2,±5/2,±9/2,±15/2,±45/2 

Let x – 1 = 0 or x = 1 

f(1) = 2(1)⁴ – 7(1)³ – 13(1)² + 63(1) – 45

= 2 – 7 – 13 + 63 – 45 

= 0 

f(1) = 0 

f(x) can be written as, 

f(x) = (x - 1) (2x³ – 5x² -18x + 45) 

or f(x) = (x - 1) g(x) …(1)

Let x – 3 = 0 or x = 3 

f(3) = 2(3)⁴ – 7(3)³ – 13(3)² + 63(3) – 45 

= 162 – 189 – 117 + 189 – 45

= 0 

f(3) = 0 

Now, we are available with 2 factors of f(x), (x – 1) and (x – 3) 

Here g(x) = 2x² (x-3) + x(x-3) -15(x-3) 

Taking (x-3) as common 

= (x-3)(2x² + x – 15) 

= (x-3)(2x²+6x – 5x -15)

= (x-3)(2x-5)(x+3)

= (x-3)(x+3)(2x-5) ….(2) 

From (1) and (2) 

f(x) =(x-1) (x-3)(x+3)(2x-5)