Problem 23 - Biocalculus Calculus for the Life Sciences

Question

$$\begin{array}{c}{\text { Suppose that } F, G, \text { and } Q \text { are polynomials and }} \\ {\frac{F(x)}{Q(x)}=\frac{G(x)}{Q(x)}}\end{array}$$ $$\begin{array}{l}{\text { for all } x \text { except when } Q(x)=0 . \text { Prove that } F(x)=G(x) \text { for }} \\ {\text { all } x .[\text {Hint} \text { : Use continuity.] }}\end{array}$$

Step-by-Step Solution

Answer

Continuity implies \( F(x) = G(x) \) for all \( x \).

Step 1: State the Given Equation and Exceptions

We start with the given equation \( \frac{F(x)}{Q(x)} = \frac{G(x)}{Q(x)} \) for all \( x \) except when \( Q(x) = 0 \). This implies wherever \( Q(x) eq 0 \), \( F(x) = G(x) \) because the fractions are identical.

Step 2: Understand Implications for Non-Zero Denominator

For all \( x \) where \( Q(x) eq 0 \), simplifying the equation results in \( F(x) = G(x) \). Thus, the equality \( F(x) = G(x) \) holds for values of \( x \) where the polynomials are defined.

Step 3: Analyze the Polynomial Properties

Since \( F(x) \) and \( G(x) \) are polynomials, they are continuous functions. This means \( F(x) = G(x) \) is continuous over its domain except potentially at the points where \( Q(x) = 0 \).

Step 4: Apply Continuity

Using continuity, since \( F(x) = G(x) \) for all points except where \( Q(x) = 0 \), and both polynomials \( F(x) \) and \( G(x) \) are defined everywhere, the equality must hold at every point, including those where \( Q(x) = 0 \).

Step 5: Conclude Equality for All x

By continuity and the fact that polynomials are defined everywhere, the equality \( F(x) = G(x) \) must hold true for all \( x \). Therefore, \( F \) and \( G \) are identical polynomials.

Key Concepts

Continuity

Continuity is a foundational concept in calculus and mathematics when dealing with functions. A function is considered continuous if, for every point in the function's domain, the limit of the function as it approaches that point is equal to the function's value at that point.
Polynomials are famously continuous functions. This continuity is significant because it ensures that if two polynomial functions are equal at almost every point in their domains, they are equal everywhere. Continuity allows us to extend the equality of polynomial functions to include points of potential undefined values, such as those where the denominator might be zero.
Simplified:

• Continuous functions have no gaps, jumps, or holes in their graphs.
• Polynomials are continuous over the entire set of real numbers.

This property of being continuous helps us conclude that two polynomial expressions, like \( F(x) \) and \( G(x) \), are equal at all possible values of \( x \). This is important in understanding how polynomials behave when divided by another polynomial, \( Q(x) \), and supports why \( F(x) = G(x) \) holds true everywhere if given \( \frac{F(x)}{Q(x)} = \frac{G(x)}{Q(x)} \).

Equality of Polynomials

The equality of polynomials means that for two polynomials to be equal, their corresponding coefficients must match. When working with polynomial functions, if the functions are equivalent at all points at which they are defined, they are inherently the same also at points where they might not be initially calculated due to a zero denominator.
Understanding polynomial equality can be broken down into easy steps:

• If two polynomials \( F(x) \) and \( G(x) \) are equal whenever the function is defined, then the expressions themselves are equal, as long as they are continuous, like polynomials are.
• This equality means that every coefficient in the polynomial expressions matches, leading to an identical polynomial and allowing us to generalize the equality to all \( x \).

This concept is particularly used to show that \( F(x) = G(x) \), given that the equality holds outside of zeros of \( Q(x) \), can be extended to all points, due to the continuity of polynomials.
The fact that polynomials equal at all points except where undefined implies they are equal at all points simply because polynomials are continuous functions.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. They have specific properties due to their polynomial components but have some unique characteristics as well.
Consider the rational function \( \frac{F(x)}{Q(x)} \):

• These functions are defined for all \( x \) except where \( Q(x) = 0 \). These undefined points are often referred to as poles or asymptotes.
• The quotient \( \frac{F(x)}{Q(x)} = \frac{G(x)}{Q(x)} \) holds true for all \( x \) except when \( Q(x) = 0 \) because at such points, the function is not defined.

When dealing with rational functions, if the numerators are determined to be equal when the denominators are not zero, we can infer that the two polynomial functions are essentially equivalent everywhere due to continuity. This is the method used in the exercise to prove \( F(x) = G(x) \).
Rational expressions are vital in demonstrating concepts of equality and continuity, as seen in polynomials and provide a unique lens on solving polynomial equations graphically and algebraically.