# Rational functions

Use the following rational function to answer the questions below.

$$f(x) = \dfrac{ \sum_{n=0}^{k} x^n }{ \sum_{n=0}^{l} x^n } = \dfrac{ 1 + x + x^2 + \ldots + x^k }{ 1 + x + x^2 + \ldots + x^l }$$

1. What do you notice about the numerator and $$f(x)$$?

1. Conjecture:
2. Example:
3. Why does this work? Will it work for all rational functions?

2. What do you notice about the denominator and $$f(x)$$?

1. Conjecture:
2. Example:
3. Why does this work? Will it work for all rational functions?

3. What do you notice when the degree of the numerator is one greater than the degree of the denominator? (e.g., k = l + 1)

1. Conjecture:
2. Example:
3. Why does this work? Will it work for all rational functions?

4. What do you notice when the degree of the numerator is two greater than the degree of the denominator? (e.g., k = l + 2)

1. Conjecture:
2. Example:
3. Why does this work? Will it work for all rational functions?

5. What do you notice when the degree of the denomonator is greater than the degree of the numerator? (e.g., k < l)

1. Conjecture:
2. Example:
3. Why does this work? Will it work for all rational functions?

Use the following rational function to answer the questions below.

$$f(x) = \dfrac{ \prod_{n=1}^{k} (x-n) }{ \prod_{n=1}^{l} (x-n) } = \dfrac{ (x - 1)(x - 2)(x - 3) \cdot \ldots \cdot (x - k) }{ (x - 1)(x - 2)(x - 3) \cdot \ldots \cdot (x - l) }$$

1. What do you notice about the numerator and $$f(x)$$?

1. Conjecture:
2. Example:
3. Why does this work? Will it work for all rational functions?

2. What do you notice about the denominator and $$f(x)$$?

1. Conjecture:
2. Example:
3. Why does this work? Will it work for all rational functions?

3. What do you notice when the degree of the numerator is one greater than the degree of the denominator? (e.g., k = l + 1)

1. Conjecture:
2. Example:
3. Why does this work? Will it work for all rational functions?

4. What do you notice when the degree of the numerator is two greater than the degree of the denominator? (e.g., k = l + 2)

1. Conjecture:
2. Example:
3. Why does this work? Will it work for all rational functions?

5. What do you notice when the degree of the denomonator is greater than the d egree of the numerator? (e.g., k < l)

1. Conjecture:
2. Example:
3. Why does this work? Will it work for all rational functions?

2016 WMC presentation