Question 212364
Find all roots and check answer 


{{{5x^3 - 8x =12x}}}


Note:  Since this is a polynomial of degree 3, it has three roots.


Step 1.  Subtract 12x from both sides to make right side of equation = 0


{{{5x^3 - 8x-12x =12x-12x=0}}}


{{{5x^3-20x=0}}}


Step 2.  Factor out 5x


{{{5x(x^2-4)=0 }}}


Step 3.  Note difference of squares {{{x^2-4=x^2-2^2}}} where


{{{x^2-2^2=(x+2)(x-2)}}}


Step 4.  The completely factored polynomial is now


{{{5x(x-2)(x+2)=0}}}


Step 5.  To make the above equation equal to zero implies each factor be set to zero to find the roots.


Step 5a.  {{{5x=0}}} implies {{{x=0}}} and check by substituting x=0 into  


{{{5x^3 - 8x =12x}}} yielding {{{0=0}}}  ok.



Step 5b.  {{{x-2=0}}} implies {{{x=2}}} and check by substituting x=2 into


{{{5x^3 - 8x =12x}}} yielding {{{5(2^3)-8(2)=12*2}}} OR  {{{40-16=24}}} OR {{{24=24}}} ok.


Step 5c.  {{{x-2=0}}} implies {{{x=2}}} and check by substituting x=2 into


{{{5x^3 - 8x =12x}}} yielding {{{5((-2)^3)-8(-2)=12*(-2)}}} OR  {{{-40+16=-24}}} OR {{{-24=-24}}} ok.


Step 6.  So the roots are: 0, 2, and -2.


Hope you understood and followed the steps.  Good luck in your studies.  Dr J


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