Question 129111
Let's factor {{{x^3-x^2-4x+4}}}


{{{x^3-x^2-4x+4}}} Start with the given expression


{{{(x^3-x^2)+(-4x+4)}}} Group like terms



{{{x^2(x-1)-4(x-1)}}} Factor out the GCF {{{x^2}}} out of the first group. Factor out the GCF {{{-4}}} out of the second group



{{{(x^2-4)(x-1)}}} Since we have the common term {{{x-1}}}, we can combine like terms


{{{(x+2)(x-2)(x-1)}}} Now factor {{{x^2-4}}} to get {{{(x+2)(x-2)}}}



So {{{x^3-x^2-4x+4}}} factors to {{{(x+2)(x-2)(x-1)}}}



Notice if we solve {{{(x+2)(x-2)(x-1)=0}}} we find the zeros {{{x=-2}}}, {{{x=1}}} and {{{x=2}}}



In order to solve {{{x^3-x^2-4x+4<0}}} we need to test some points. So let's pick a point that is less than {{{x=-2}}}


So let's test {{{x=-3}}}



{{{x^3-x^2-4x+4<0}}} Start with the given inequality



{{{(-3)^3-(-3)^2-4(-3)+4<0}}} Plug in {{{x=-3}}} 


{{{-20<0}}} Simplify. Since this inequality is true, any value that is less than  {{{x=-2}}} will satisfy the inequality.




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Now let's test a value that is in between {{{x=-2}}} and {{{x=1}}}



So let's test {{{x=0}}}



{{{x^3-x^2-4x+4<0}}} Start with the given inequality



{{{(0)^3-(0)^2-4(0)+4<0}}} Plug in {{{x=0}}} 


{{{4<0}}} Simplify. Since this inequality is <b>not</b> true, this means that the interval [-2,1] is <b>not</b> in the solution set.




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Now let's test a value that is in between {{{x=1}}} and {{{x=2}}}



So let's test {{{x=1.5}}}



{{{x^3-x^2-4x+4<0}}} Start with the given inequality



{{{(1.5)^3-(1.5)^2-4(1.5)+4<0}}} Plug in {{{x=1.5}}} 


{{{-0.875<0}}} Simplify. Since this inequality is true, any value that is in between x=1 and x=2 will satisfy the inequality.



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Now let's test a value that is greater than {{{x=2}}}



So let's test {{{x=3}}}



{{{x^3-x^2-4x+4<0}}} Start with the given inequality



{{{(3)^3-(3)^2-4(3)+4<0}}} Plug in {{{x=3}}} 


{{{10<0}}} Simplify. Since this inequality is <b>not</b> true, this means that any value greater than x=2 will <b>not</b> satisfy the inequality




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Answer:



So the solution set is


*[Tex \LARGE \left(\infty,-2\right)\cup\left(1,2\right)]




Notice if we graph {{{y=x^3-x^2-4x+4}}}, we can visually verify our answer.



{{{ graph( 500, 500, -10, 10, -10, 10, x^3-x^2-4x+4) }}}