A few days ago
If x^2=x+1 then what is x^3?
This question was on an exam I recently took and I couldn’t figure it out. For some reason this just has me stumped. There were multiple choice options–which I can’t remember–but none of the options seemed reasonable at all. Please provide the answer and how you came up with the answer. Thanks!
Top 2 Answers
A few days ago
Favorite Answer
First of all solve x^2 – x – 1 = 0 giving x = (1 ± sqrt(5))/2. Take the plus sign for the moment.
Now x^2 = 2x + 1 = (3 + sqrt(5))/2 (you don’t even need this)
And x^3 = x(x^2) = x(x + 1) = x^2 + x = x + 1 + x = 2x + 1
and so x^3 = 2 + sqrt(5). (!!)
If you take the negative sign on the square root, you get x^3 = 2 – sqrt(5).
The key is that because of the equation given, all second powers can be reduced to first powers. So every power of x can be expressed as ax+b for some integers a and b. Weird but true!
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A few days ago
x^3= x^2+x
If x^2=x+1, then multiply both sides by “x”
x(x^2)= x(x+1)
Distribute the x as desired.
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