Question

ready
topic: connecting features of polynomials and rational functions
find the roots and domain for each function.

1. $f(x) = (x + 5)(x - 2)(x - 7)$
2. $g(x) = x^2 + 7x + 6$
3. $k(x) = \frac{1}{(x+5)(x-2)(x-7)}$
4. $h(x) = \frac{1}{(x^2 + 7x + 6)}$
5. make a conjecture that compares the domain of a polynomial with the domain of the reciprocal of the polynomial. (note that the reciprocal of a polynomial is a rational function.)
6. do the roots of the polynomial tell you anything about the graph of the reciprocal of the polynomial? explain.
7. find the y-intercept for #1 and #2. what is the y-intercept for #3 and #4?

set
topic: distinguishing between proper and improper rational functions.
determine if each of the following is a proper or an improper rational function.

1. $f(x) = \frac{x^3 + 3x^2 + 7}{7x^2 - 2x + 1}$
2. $f(x) = x^3 - 5x^2 - 4$
3. $f(x) = \frac{3x^2 - 2x + 7}{x^5 - 5}$

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Explanation:

Step1: Recall the definition of rational functions

A rational function is of the form \( \frac{N(x)}{D(x)} \), where \( N(x) \) is the numerator polynomial and \( D(x) \) is the denominator polynomial. A rational function is proper if the degree of the numerator \( \text{deg}(N(x)) \) is less than the degree of the denominator \( \text{deg}(D(x)) \). It is improper if \( \text{deg}(N(x)) \geq \text{deg}(D(x)) \).

Step2: Analyze function 8

For \( f(x)=\frac{x^{3}+3x^{2}+7}{7x^{2}-2x + 1} \):

• The degree of the numerator \( N(x)=x^{3}+3x^{2}+7 \) is \( 3 \) (the highest power of \( x \) is \( 3 \)).
• The degree of the denominator \( D(x)=7x^{2}-2x + 1 \) is \( 2 \).
• Since \( 3>2 \) ( \( \text{deg}(N(x))>\text{deg}(D(x)) \) ), the function is improper.

Step3: Analyze function 9

For \( f(x)=x^{3}-5x^{2}-4 \), we can rewrite it as \( \frac{x^{3}-5x^{2}-4}{1} \) (since any polynomial can be considered as a rational function with denominator \( 1 \)):

• The degree of the numerator \( N(x)=x^{3}-5x^{2}-4 \) is \( 3 \).
• The degree of the denominator \( D(x) = 1 \) (since \( 1=x^{0} \), degree is \( 0 \)). Wait, no, actually, when the denominator is \( 1 \), we can think of it as a special case. But the rule still applies: \( \text{deg}(N(x))=3 \), \( \text{deg}(D(x)) = 0 \). Since \( 3\geq0 \), the function is improper (in fact, all polynomials are improper rational functions when written in the form \( \frac{\text{polynomial}}{1} \) because the degree of the numerator is at least \( 0 \) and the degree of the denominator is \( 0 \), and for non - zero polynomials, the degree of the numerator is greater than or equal to \( 0 \)).

Step4: Analyze function 10

For \( f(x)=\frac{3x^{2}-2x + 7}{x^{5}-5} \):

• The degree of the numerator \( N(x)=3x^{2}-2x + 7 \) is \( 2 \).
• The degree of the denominator \( D(x)=x^{5}-5 \) is \( 5 \).
• Since \( 2<5 \) ( \( \text{deg}(N(x))<\text{deg}(D(x)) \) ), the function is proper.

Answer:

1. Improper (because the degree of the numerator (3) is greater than the degree of the denominator (2))
2. Improper (because the degree of the numerator (3) is greater than the degree of the denominator (0, since the denominator is 1))
3. Proper (because the degree of the numerator (2) is less than the degree of the denominator (5))