SOLUTION: How many real numbers are there such that the 5th power of the number is the sum of the 4th and 3rd powers of the number? a) 1 b) 2 c) 3 d) 5 e) none of these

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Question 283083: How many real numbers are there such that the 5th power of the number is the sum of the 4th and 3rd powers of the number?
a) 1 b) 2 c) 3 d) 5 e) none of these
Answer by nabla(475) About Me  (Show Source):
You can put this solution on YOUR website!
So, we want:
x%5E5=x%5E4%2Bx%5E3
In other words we are asking when
x%5E5-x%5E4-x%5E3=0?
In even more words we are asking when
x%5E3%28x%5E2-x-1%29=0?
We note that x=0 certainly makes the statement true. Now, x%5E2-x-1 will give two real (irrational) roots. So, strictly speaking there are three real numbers that meet the criterion: {0, 1/2 (1 - Sqrt[5]), 1/2 (1 + Sqrt[5]).
However, one should note that x=0 occurs as a root three times.