simplify (1+t^4)(1+t^6)(1-t^10)/(1-t^2)

asked by guest
on Jan 18, 2025 at 2:14 am



You asked:

Simplify the expression: \(\frac{\left(1 + {t}^{4}\right) \left(1 + {t}^{6}\right) \left(1 - {t}^{10}\right)}{1 - {t}^{2}}\)

MathBot Answer:

Simplified:

\[ \frac{\left(1 + {t}^{4}\right) \left(1 + {t}^{6}\right) \left(1 - {t}^{10}\right)}{1 - {t}^{2}} = \frac{\left(t^{4} + 1\right) \left(t^{6} + 1\right) \left(t^{10} - 1\right)}{t^{2} - 1} \]

Expanded:

\[ \frac{\left(1 + {t}^{4}\right) \left(1 + {t}^{6}\right) \left(1 - {t}^{10}\right)}{1 - {t}^{2}} = - \frac{t^{20}}{1 - t^{2}} - \frac{t^{16}}{1 - t^{2}} - \frac{t^{14}}{1 - t^{2}} + \frac{t^{6}}{1 - t^{2}} + \frac{t^{4}}{1 - t^{2}} + \frac{1}{1 - t^{2}} \]

Factored:

\[ \frac{\left(1 + {t}^{4}\right) \left(1 + {t}^{6}\right) \left(1 - {t}^{10}\right)}{1 - {t}^{2}} = \left(t^{2} + 1\right) \left(t^{4} + 1\right) \left(t^{4} - t^{2} + 1\right) \left(t^{4} - t^{3} + t^{2} - t + 1\right) \left(t^{4} + t^{3} + t^{2} + t + 1\right) \]


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