If \(x+\frac{1}{x}=5\), then \(\frac{x^{3}-5x^{2}+5x}{x^{2}+1}\) is equal to:
\(\frac{4}{5}\)
To solve the expression \(\frac{x^{3}-5x^{2}+5x}{x^{2}+1}\) given that \(x+\frac{1}{x}=5\), we will use algebraic identities and substitution.
Firstly, let's manipulate the given equation:
Given:
\(x + \frac{1}{x} = 5\)
We square both sides to find related expressions:
\(\left( x + \frac{1}{x} \right)^2 = 5^2\)
\(x^2 + 2 + \frac{1}{x^2} = 25\)
\(x^2 + \frac{1}{x^2} = 23\)
Now, let's express \(\frac{x^3 - 5x^2 + 5x}{x^2 + 1}\) step-by-step:
The expression \(x^3\) can be found using:
\(x^3 = x \cdot x^2\)
From \((x+\frac{1}{x})^2\), we know:
\(x^2 = 5x - 1\)
Now:
\(x^3 = x \cdot (5x - 1) = 5x^2 - x\)
Now substitute \(x^3\), \(5x^2\), and \(5x\) directly in the main expression:
Expression:
\(\frac{(5x^2 - x) - 5x^2 + 5x}{x^2 + 1}\)
\(= \frac{-x + 5x}{x^2 + 1}\)
\(= \frac{4x}{x^2 + 1}\)
Finally, express \(\frac{x}{x^2 + 1}\):
If \(x+\frac{1}{x}=5\), multiplying both sides by \(\frac{1}{x}\) gives:
\(1 + \frac{1}{x^2} = \frac{5}{x}\)
Thus, we simplify:
\(\frac{4x}{x^2 + 1} = \frac{4x}{23}\)
Calculating further
Using:
\(\frac{4}{5}\)
The correct answer is therefore:
\(\frac{4}{5}\)
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