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'''Roots of Polynomials'''<br>
 
'''Roots of Polynomials'''<br>
:::'''$\rightarrow$  Intermediate Value Theorem (IVT) Conjecture''' <br>
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:::'''-> Intermediate Value Theorem (IVT) Conjecture''' <br>
:::'''$\rightarrow$  Bisection Approximation Method $\rightarrow$  Investigating Resulting Sequences''' <br>
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:::'''-> Bisection Approximation Method -> Investigating Resulting Sequences''' <br>
:::'''$\rightarrow$  [Endpoint Option 1] Prove Increasing Unbounded Sequences "Tend to Infinity"''' <br>
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:::'''-> [Endpoint Option 1] Prove Increasing Unbounded Sequences "Tend to Infinity"''' <br>
:::'''$\rightarrow$  Monotone Convergence Conjecture $\rightarrow$  Define Convergence $\rightarrow$  Define Least Upper Bound '''<br>
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:::'''-> Monotone Convergence Conjecture -> Define Convergence -> Define Least Upper Bound '''<br>
:::'''$\rightarrow$  [Endpoint Option 2] Prove Monotone Convergence <br>
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:::'''-> [Endpoint Option 2] Prove Monotone Convergence <br>
:::'''$\rightarrow$  Define Convergence / Sequential Characterization of Continuity $\rightarrow$  [Endpoint Option 3] Prove IVT'''<br>
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:::'''-> Define Convergence / Sequential Characterization of Continuity -> [Endpoint Option 3] Prove IVT'''<br>
  
 
# Students begin by considering questions about whether polynomials of odd/even degree are guaranteed to have real roots.  
 
# Students begin by considering questions about whether polynomials of odd/even degree are guaranteed to have real roots.  

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