Question

Using the Factor Theorem, show that:
(x - 2) is a factor of x³-2x² - 9x + 18.
Hence, factorise the expression
X³- 2x²- 9x + 18 completely.​

Answer 1

Since x - 2 is a factor of the given equation,

x - 2 = 0

⇒x = 2

p(x) = x³ - 2x² - 9x + 18

Substitute value of x as 2

⇒p(2) = 2³ - 2(2)² - 9(2) + 18

⇒p(2) = 8 - 8 - 18 + 18

⇒ p(2) = 0

So, 2 is a factor of x³ - 2x² - 9x + 18

For factorising the given polynomial, divide the polynomial by x - 2

Division is given in attachment

x³ - 2x² - 9x + 18 = (x - 2)(x² - 9)

We can split x² - 9 by using

a² - b² = (a + b)(a - b)

⇒ x² - 9 = (x + 3)(x - 3)

Factors:

1. (x - 2)
2. (x + 3)
3. (x - 3)