Determine the factors of the following polynomial: x³-2x²-5x+6

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

Step-by-step explanation:

The leading coefficient is 1, and the constant is 6. Using the rational root theorem, the possible real roots are:

±1, ±2, ±3, ±6

Trying them all:

(-1) ³ - 2 (-1) ² - 5 (-1) + 6 = 8

(1) ³ - 2 (1) ² - 5 (1) + 6 = 0

(-2) ³ - 2 (-2) ² - 5 (-2) + 6 = 0

(2) ³ - 2 (2) ² - 5 (2) + 6 = - 4

(-3) ³ - 2 (-3) ² - 5 (-3) + 6 = - 24

(3) ³ - 2 (3) ² - 5 (3) + 6 = 0

(-6) ³ - 2 (-6) ² - 5 (-6) + 6 = - 252

(6) ³ - 2 (6) ² - 5 (6) + 6 = 120

So - 2, 1, and 3 are all roots. Since the equation is cubic, there can only be three roots, so there are no irrational roots to find.

Therefore:

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