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)