Factorize the equation: x³ - 3x² - 9x - 5.

We will find the factors of the expression x³ - 3x² - 9x - 5

Answer: The equation can be factorized as: x³ - 3x² - 9x - 5 = (x+1)² (x-5)

We will find one of the zeros of the expression x³ – 3x² – 9x – 5 and use long division to find the other zeros.

Explanation:

Substitute 5 for x in x³ - 3x² - 9x - 5.

5³ - 3(5)² - 9(5) - 5 = 125 - 75 - 45 - 5 = 0

This means, by factor theorem, that (x - 5) is a factor of x³ - 3x² - 9x - 5.

Divide x³ - 3x² - 9x - 5 by x - 5 using long division of polynomials:

Here, the quotient is x² + 2x + 1.

This quadratic expression can be factorized as follows:

x² + 2x + 1 = x² + x + x + 1

= x (x + 1) + 1 (x + 1)

= (x + 1)(x + 1)

= (x + 1)²

This means x³ - 3x² - 9x - 5 = (x - 5) (x² + 2x + 1) = (x - 5) (x + 1)²

Therefore, the factorization of x³ - 3x² - 9x - 5 is (x - 5) (x + 1)².