Question

Using factor theorem, factorize each of the following polynomial:

2x⁴ - 7x³ - 13x² + 63x - 45

Answer 1

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

The factors of the constant term – 45 are +1,+3,+5,+9,+15 and + 45

The factor of the coefficient of x 4 is 2. Hence, possible rational roots of f (x) are:

We have,

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

= 2 – 7 – 13 + 63 – 45

= 0

And,

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

= 162 – 189 – 117 + 189 – 45

= 0

So, (x – 1) and (x + 3) are the factors of f (x)

(x – 1) (x + 3) is also a factor of f (x)

Let us now divide

f (x) = 2x⁴ - 7x³ - 13x² + 63x - 45 by (x² – 4x + 3) to get the other factors of f (x)

Using long division method, we get

2x⁴ - 7x³ - 13x² + 63x - 45 = (x² – 4x + 3) (2x² + x – 15)

2x⁴ - 7x³ - 13x² + 63x - 45 = (x – 1) (x – 3) (2x² + x – 15)

Now,

2x² + x – 15 = 2x² + 6x – 5x – 15

= 2x (x + 3) – 5 (x + 3)

= (2x – 5) (x + 3)

Hence, 

2x⁴ - 7x³ - 13x² + 63x - 45 = (x – 1) (x – 3) (x + 3) (2x – 5)