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)

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step-by-step explanation:

-12 > a

step-by-step explanation:

-11 > 1 + a

subtract 1 from each side

-11-1 > 1-1+a

-12 > a