Answer: A) x+3
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Step-by-step explanation:
We can use the special case of the remainder theorem, aka the factor theorem, which says:
If p(k) = 0, then x-k is a factor of p(x).
For choice A, we have x-k = x+3 = x-(-3) to show that k = -3
Plug that k value into the given expression
x^3 - 3x^2 - 4x + 12
(-3)^3 - 3(-3)^2 - 4(-3) + 12
-30
We don't get 0 which means x+3 is not a factor of the expression above. Therefore, choice A is the final answer.
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If we tried something like choice B, then k = 3 leads us to...
x^3 - 3x^2 - 4x + 12
(3)^3 - 3(3)^2 - 4(3) + 12
0
which points us to (x-3) being a factor of the original expression. The same happens with choices C and D as well. This allows us to rule out choices B, C and D.
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A quick alternative is to graph x^3 - 3x^2 - 4x + 12 using software like Desmos or GeoGebra. I posted the graph below. The graph shows that the roots are x = 3, x = 2, x = -2. Those roots mentioned lead to factors x-3, x-2, x+2 in that order (which correspond to choices B,C,D in the same order).
In short, the roots tell us the factors fairly quickly, which allows us to cross off the non-answers.