Polynomial Division
If x − 1 2 = x 1 for x ∈ R + , find
x 4 − 1 0 x 3 − 1 5 x 2 − 1 2 2 x + 2 0 1 2
The answer is 2022.
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2 solutions
This is the most efficient way to do this!
Interesting approach haha...
If x − 1 2 = x 1 , multiply both sides of the equation with x and move all terms to the left side of the equation:
x 2 − 1 2 x − 1 = 0
Hence, by long division,
x 4 − 1 0 x 3 − 1 5 x 2 − 1 2 2 x + 2 0 1 2 = ( x 2 − 1 2 x − 1 ) ( x 2 + 2 x + 1 0 ) + 2 0 2 2 = ( 0 ) ( x 2 + 2 x + 1 0 ) + 2 0 2 2 = 2 0 2 2
For completeness, it's better to show that there exists a positive solution to the equation x − 1 2 = 1 / x
Agreed. @Pi Han Goh
Given that x − 1 2 = x 1 , ⟹ x 2 − 1 2 x = 1 . Then we have
X = x 4 − 1 0 x 3 − 1 5 x 2 − 1 2 2 x + 2 0 1 2 = x 2 ( x 2 − 1 2 x ) + 2 x 3 − 1 5 x 2 − 1 2 2 x + 2 0 1 2 = x 2 + 2 x 3 − 1 5 x 2 − 1 2 2 x + 2 0 1 2 = 2 x 3 − 1 4 x 2 − 1 2 2 x + 2 0 1 2 = 2 x ( x 2 − 1 2 x ) + 1 0 x 2 − 1 2 2 x + 2 0 1 2 = 2 x + 1 0 x 2 − 1 2 2 x + 2 0 1 2 = 1 0 x 2 − 1 2 0 x + 2 0 1 2 = 1 0 ( x 2 − 1 2 x ) + 2 0 1 2 = 1 0 + 2 0 1 2 = 2 0 2 2