Question 178968
{{{(x + 3)^5 - 4(x + 3)^3 = 0}}} Start with the given equation.



Let {{{z=x+3}}}



{{{z^5 - 4z^3 = 0}}} Replace each "x+3" with "z".



{{{z^3(z^2 - 4) = 0}}} Factor out the GCF {{{z^3}}} (note: you can avoid the substitution, but factoring is much easier to see)



{{{z^3(z+2)(z-2) = 0}}} Factor {{{z^2-1}}} into {{{(z+2)(z-2)}}} by use of the difference of squares formula



{{{z^3=0}}}, {{{z+2=0}}}, or {{{z-2=0}}} Set each factor equal to zero



{{{z=0}}}, {{{z=-2}}}, or {{{z=2}}} Solve for "z" in each case



{{{x+3=0}}}, {{{x+3=-2}}}, or {{{x+3=2}}} Plug in {{{z=x+3}}} (remember, we want to solve for "x")



{{{x=0-3}}}, {{{x=-2-3}}}, or {{{x=2-3}}} Subtract 3 from both sides (for each equation)



{{{x=-3}}}, {{{x=-5}}}, or {{{x=-1}}} Combine like terms.



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Answer:



So the solutions are {{{x=-3}}}, {{{x=-5}}}, or {{{x=-1}}}




Note: if you plug in any solution into the given equation {{{(x + 3)^5 - 4(x + 3)^3 = 0}}} and simplify, you should get {{{0=0}}} . If you do this to all of the solutions, then this will verify the answer.