Question

Use factor theorem to factorise the following polynominals completely.

x³ + 2x² – 5x – 6

Answer 1

Let f(x) = x³ + 2x² – 5x – 6

Factors of (∵ 6 = ± 1 : ± 2, ± 3, ±)

Let x = –1, then

f(–1) = (–1)³ + 2(–1)² – 5(–1) – 6

= –1 + 2(1) + 5 – 6

= –1 + 2 + 5 – 6

= 7 – 7

= 0

∵ f(–1) = 0

∴ x + 1 is a factor of f(x).

Similarly, (x - 2) and (x + 3) are the factors of f(x).

Since f(x) is a polynomial of degree 3.

So, it can not have more than three linear factors.

∴ f(x) = k(x + 1) (x - 2)(x + 3)           ...(1) 

⇒ x³ + 2x² - 5x - 6

= k(x + 1)(x - 2)(x + 3) 

Putting x = 0 on both sides, we get

-6 = k(1)(-2)(3)

⇒ k = 1

Putting k = 1 in equation (1), we get

f(x) = (x + 1)(x + 3)

Hence, x³ + 2x² - 5x - 6 = (x + 1)(x - 2)(x + 3)