\(x^{4}\)+\(x^{3}\)+\(x^{2}\)+x+1を2次式の積で表しなさい。
という問題です。どなたか解き方を教えてください。お願いします。
\(x^{4}\)+\(x^{3}\)+\(x^{2}\)+x+1を2次式の積で表しなさい。
という問題です。どなたか解き方を教えてください。お願いします。
\(x^{4}\) + \(x^{3}\) + \(x^{2}\) + x + 1
= \(x^{2}\)(\(x^{2}\) + x + 1 + \(\frac{1}{x}\) + 1/\(x^{2}\))
= \(x^{2}\)((\(x^{2}\) + 1/\(x^{2}\)) + (x + \(\frac{1}{x}\)) + 1)
= \(x^{2}\)((x + \(\frac{1}{x}\)\()^{2}\) - 2 + (x + \(\frac{1}{x}\)) + 1)
= \(x^{2}\)((x + \(\frac{1}{x}\)\()^{2}\) + (x + \(\frac{1}{x}\)) - 1)
= \(x^{2}\)((x + \(\frac{1}{x}\) +\(\frac{1}{2}\)\()^{2}\) - \(\frac{5}{4}\))
= \(x^{2}\)(x + \(\frac{1}{x}\) + \(\frac{1}{2}\) - (\(\sqrt{\quad}\)5)/2)(x + \(\frac{1}{x}\) + \(\frac{1}{2}\) + (\(\sqrt{\quad}\)5)/2)
= (\(x^{2}\) + 1 + (1 - \(\sqrt{\quad}\)5)\(\frac{x}{2}\))(\(x^{2}\) + 1 + (1 + \(\sqrt{\quad}\)5)\(\frac{x}{2}\))
= (\(x^{2}\) + (1 - \(\sqrt{\quad}\)5)\(\frac{x}{2}\) + 1)(\(x^{2}\) + (1 + \(\sqrt{\quad}\)5)\(\frac{x}{2}\) + 1).