Question 1168599
 Use the Rational Zero Theorem to list all possible rational zeros for the given function

 {{{f(x) = -2x^3 + 3x^2 -4x + 8}}}


Since all coefficients are integers, we can apply the rational zeros theorem.

The trailing coefficient (coefficient of the constant term) is {{{8}}}.

Find its factors (with plus and minus): ±{{{1}}},±{{{2}}},±{{{4}}},±{{{8}}}. These are the possible values for {{{p}}}.

The leading coefficient (coefficient of the term with the highest degree) is {{{-2}}}.

Find its factors (with plus and minus): ±{{{1}}},±{{{2}}}. These are the possible values for {{{q}}}.

Find all possible values of {{{p/q}}}:
 ±{{{1/1}}},±{{{1/2}}},±{{{2/1}}},±{{{2/2}}},±{{{4/1}}},±{{{4/2}}},±{{{8/1}}},±{{{8/2}}}.

Simplify and remove duplicates (if any), these are {{{highlight(possible)}}} rational roots: ±{{{1}}},±{{{1/2}}},±{{{2}}},±{{{4}}},±{{{8}}}



Next, check the possible roots: 
if {{{highlight(a)}}} is a {{{root}}} of the polynomial {{{f(x)}}}, the remainder from the division of {{{f(x)}}} by {{{highlight(x-a)}}} should equal {{{highlight(0)}}}.

Check {{{1}}}: divide {{{-2x^3+3x^2-4x+8}}} by {{{x-1}}}.The quotient is {{{-2x^2+x-3}}} and the remainder is {{{5}}} (use the synthetic division calculator to see the steps).

Check {{{-1}}}: divide {{{-2x^3+3x^2-4x+8}}} by {{{x+1}}}.The quotient is {{{-2x^2+5x-9}}} and the remainder is {{{17}}} (use the synthetic division calculator to see the steps).

Check {{{1/2}}}: divide {{{-2x^3+3x^2-4x+8}}} by {{{x-1/2}}}.The quotient is {{{-2x^2+2x−3}}} and the remainder is {{{13/2}}} (use the synthetic division calculator to see the steps).

Check {{{-1/2}}}: divide {{{-2x^3+3x^2-4x+8}}} by {{{x+1/2}}}.The quotient is {{{-2x^2+4x-6}}} and the remainder is {{{11}}} (use the synthetic division calculator to see the steps).

Check {{{2}}}: divide {{{-2x^3+3x^2-4x+8}}} by {{{x-2}}}.The quotient is {{{-2x^2-x-6}}} and the remainder is {{{-4}}} (use the synthetic division calculator to see the steps).

Check {{{-2}}}: divide {{{-2x^3+3x^2-4x+8}}} by {{{x+2}}}.The quotient is {{{-2x^2+7x-18}}} and the remainder is {{{44}}} (use the synthetic division calculator to see the steps).

Check {{{4}}}: divide {{{-2x^3+3x^2-4x+8}}} by {{{x-4}}}.The quotient is {{{-2x^2-5x-24}}} and the remainder is {{{-88}}} (use the synthetic division calculator to see the steps).

Check {{{-4}}}: divide {{{-2x^3+3x^2-4x+8}}} by {{{x+4}}}.The quotient is {{{-2x^2+11x-48}}} and the remainder is {{{200 }}}(use the synthetic division calculator to see the steps).

Check {{{8}}}: divide {{{-2x^3+3x^2-4x+8}}} by {{{x-8}}}.The quotient is {{{-2x^2-13x-108}}} and the remainder is {{{-856}}} (use the synthetic division calculator to see the steps).

Check {{{-8}}}: divide {{{-2x^3+3x^2-4x+8}}} by {{{x+8}}}.The quotient is {{{-2x^2+19x-156}}} and the remainder is {{{1256}}} (use the synthetic division calculator to see the steps).


so, none of these possible rational roots  ±{{{1}}},±{{{1/2}}},±{{{2}}},±{{{4}}},±{{{8}}} are real roots of given function because long division by them does not gives us reminder {{{highlight(0)}}}