Question
Consider the polynomial function \( f(t)=t^{9}+2 t^{6}+3 t^{2}+4 \) As \( t \rightarrow-\infty, f(t) \rightarrow \) ? As \( t \rightarrow \infty, f(t) \rightarrow \) ?
Ask by Lyons Sandoval. in the United States
Mar 17,2025
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Answer
As \( t \) approaches negative infinity, \( f(t) \) approaches negative infinity. As \( t \) approaches positive infinity, \( f(t) \) approaches positive infinity.
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Beyond the Answer
As \( t \rightarrow -\infty \), the leading term \( t^9 \) dominates the behavior of the polynomial function. Since the degree of the polynomial is odd and the coefficient of the leading term is positive, \( f(t) \) approaches \( -\infty \). As \( t \rightarrow \infty \), again the leading term \( t^9 \) dictates the behavior of the polynomial. With a positive coefficient for the leading term, \( f(t) \) approaches \( +\infty \).