<PRE><B>Question 656264</B>
Factors of 6: 1, 2, 3, 6, -1, -2, -3, -6


Factors of 1: 1, -1


*** Note: Include negative factors as well ***


Divide all the factors of 6 by the factors of 1 to get the potential rational roots of x^3-2x^2-5x+6: 


1, -1, 2, -2, 3, -3, 6, -6


Note: there are 8 possible rational roots (listed above) for x^3-2x^2-5x+6.



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Let&#39;s check all the possible roots to see if there are any actual rational roots.



Check to see if x = 1 is a root for x^3-2x^2-5x+6:


(1)^3-2(1)^2-5(1)+6 = 0


Since the result is 0, x = 1 is a root for x^3-2x^2-5x+6.



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Check to see if x = -1 is a root for x^3-2x^2-5x+6:


(-1)^3-2(-1)^2-5(-1)+6 = 8


Since the result is NOT 0, x = -1 is NOT a root for x^3-2x^2-5x+6.



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Check to see if x = 2 is a root for x^3-2x^2-5x+6:


(2)^3-2(2)^2-5(2)+6 = -4


Since the result is NOT 0, x = 2 is NOT a root for x^3-2x^2-5x+6.



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Check to see if x = -2 is a root for x^3-2x^2-5x+6:


(-2)^3-2(-2)^2-5(-2)+6 = 0


Since the result is 0, x = -2 is a root for x^3-2x^2-5x+6.



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Check to see if x = 3 is a root for x^3-2x^2-5x+6:


(3)^3-2(3)^2-5(3)+6 = 0


Since the result is 0, x = 3 is a root for x^3-2x^2-5x+6.



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Check to see if x = -3 is a root for x^3-2x^2-5x+6:


(-3)^3-2(-3)^2-5(-3)+6 = -24


Since the result is NOT 0, x = -3 is NOT a root for x^3-2x^2-5x+6.



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Check to see if x = 6 is a root for x^3-2x^2-5x+6:


(6)^3-2(6)^2-5(6)+6 = 120


Since the result is NOT 0, x = 6 is NOT a root for x^3-2x^2-5x+6.



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Check to see if x = -6 is a root for x^3-2x^2-5x+6:


(-6)^3-2(-6)^2-5(-6)+6 = -252


Since the result is NOT 0, x = -6 is NOT a root for x^3-2x^2-5x+6.



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Answer:


The polynomial x^3-2x^2-5x+6 has 3 rational roots and they are: x = 1, x = -2, x = 3, 


So the rational factors are: (x - 1), (x + 2) and (x - 3)


This means that x^3-2x^2-5x+6 factors to (x - 1)(x + 2)(x - 3)</PRE>