Write as a product of linear factors:

f (x) = x^5 - 6x^4 - x^3 + 32x^2 - 6x - 20

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

Step-by-step explanation:

Use rational root theorem to check for rational roots. Possible roots are:

±1, ±2, ±4, ±5, ±10, ±20

Plug each one into f (x). We find that three of the possibilities are roots: x = - 2, x = 1, and x = 5.

f (x) is a 5th order polynomial, so there are 5 total roots. To find the other 2, we divide f (x) by the product of (x + 2) (x - 1) (x - 5). That product is:

(x + 2) (x - 1) (x - 5)

= (x² + x - 2) (x - 5)

= x³ + x² - 2x - 5x² - 5x + 10

= x³ - 4x² - 7x + 10

Using long division, we find that the other factor of f (x) is x² - 2x - 2. We can find the roots of this with quadratic formula:

x = [ - b ± √ (b² - 4ac) ] / 2a

x = [ 2 ± √ (4 - 4 (1) (-2)) ] / 2

x = (2 ± √12) / 2

x = (2 ± 2√3) / 2

x = 1 ± √3

So f (x) can be written as:

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