Question 166375
Let's graph the function.
{{{ graph( 300, 300, -5, 5, -5, 5, 2x^3 - x^2 - 12x + 6) }}}
Looks like x=0.5 is a zero.
{{{2x^3 - x^2 - 12x + 6=2(0.5)^3-(0.5)^2-12(0.5)+6}}}
{{{2x^3 - x^2 - 12x + 6=(0.5)^2-(0.5)^2-6+6}}}
{{{2x^3 - x^2 - 12x + 6=0}}}
Even though x=2.5 and x=-2.5 look like zeros, they don't solve.
Let's use synthetic divsion to factor out (x-0.5) and get the quadratic equation that's left over.
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Showing synthetic division is a little difficult.
The first part will show the factor (left hand column), then the factor times the divisor,
then the next line will show the subtraction.
Repeat.
Hopefully it makes sense.
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(x-0.5)|2x^3-x^2-12x+6 
2x^2:....2x^3-x^2
.................................-12x+6
-12:..........................-12x+6



{{{(2x^3-x^2-12x+6)/(x-0.5)=2x^2-12}}}
{{{(2x^3-x^2-12x+6)=(x-0.5)*(2x^2-12)}}}
Then
{{{2x^2-12=0}}}
{{{x^2=6}}}
{{{x=0 +- sqrt(6)}}}
The three zeros are {{{1/2}}}, {{{sqrt(6)}}}, {{{-sqrt(6)}}}.