SOLUTION: Find all real solutions of the following equation. (x + 3)5 - 4(x + 3)3 = 0 x =__ (smallest solution) x =___ x = ___ (largest solution) i cant seem to figure out how to do

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find all real solutions of the following equation. (x + 3)5 - 4(x + 3)3 = 0 x =__ (smallest solution) x =___ x = ___ (largest solution) i cant seem to figure out how to do      Log On


   

Question 178968: Find all real solutions of the following equation.
(x + 3)5 - 4(x + 3)3 = 0
x =__ (smallest solution)
x =___
x = ___ (largest solution)
i cant seem to figure out how to do this! Help! thanks
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%28x+%2B+3%29%5E5+-+4%28x+%2B+3%29%5E3+=+0 Start with the given equation.


Let z=x%2B3


z%5E5+-+4z%5E3+=+0 Replace each "x+3" with "z".


z%5E3%28z%5E2+-+4%29+=+0 Factor out the GCF z%5E3 (note: you can avoid the substitution, but factoring is much easier to see)


z%5E3%28z%2B2%29%28z-2%29+=+0 Factor z%5E2-1 into %28z%2B2%29%28z-2%29 by use of the difference of squares formula


z%5E3=0, z%2B2=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%2B3=0, x%2B3=-2, or x%2B3=2 Plug in z=x%2B3 (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 %28x+%2B+3%29%5E5+-+4%28x+%2B+3%29%5E3+=+0 and simplify, you should get 0=0 . If you do this to all of the solutions, then this will verify the answer.