Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

A real zero of the numerator of a rational function \(f\) is \(x=c .\) Must \(x=c\) also be a zero of \(f ?\) Explain.

Step-by-Step Solution

Answer

Yes, if \(x = c\) is a zero of the numerator of a rational function \(f\), then it must also be a zero of \(f\), provided that \(c\) is not a zero of the denominator.

Step 1 – Understanding Rational Functions

Recall that a rational function is defined as \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomial functions. The zero(s) of \(P(x)\), or the numerator, occur at the values of \(x\) for which \(P(x) = 0\). Similarly, the zero(s) of \(Q(x)\), or the denominator, occur at the values of \(x\) for which \(Q(x) = 0\).

Step 2 – Considering Zeros of the Numerator

If \(x = c\) is a zero of \(P(x)\), that means \(P(c) = 0\). Substituting into the rational function yields \(f(c) = \frac{P(c)}{Q(c)} = \frac{0}{Q(c)}\).

Step 3 – Analyzing the Result

From the above, it can be seen that the function \(f(x)\) at \(x = c\) will be zero provided that \(Q(c)\) is not equal to zero. Because division by zero is undefined, it is important that we establish \(Q(c) \neq 0\). If this is the case, then \(f(c)\) is indeed zero. So, if \(x = c\) is a zero of the numerator of the function, then it is also a zero of \(f\), provided that \(c\) is not a zero of the denominator.

Key Concepts

Polynomials

Polynomials are a fundamental concept in algebra. They are expressions composed of variables and coefficients, constructed using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A standard polynomial is expressed in the form:
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \] where \(a_n\), \(a_{n-1}\), ..., \(a_0\) are coefficients, and \(n\) is a non-negative integer representing the degree of the polynomial. The degree is determined by the highest power of \(x\).

• Polynomials can model various curves and can be used in equations to determine relationships between variables.
• The zeros of a polynomial are the values of \(x\) that satisfy \(P(x) = 0\). These zeros are critical in graphing and analyzing functions.

Understanding polynomials is crucial for exploring more complex mathematical functions such as rational functions.

Numerator and Denominator

In the realm of rational functions, you frequently encounter the terms "numerator" and "denominator." A rational function is expressed as the ratio of two polynomials:
\[ f(x) = \frac{P(x)}{Q(x)} \] where \(P(x)\) is the numerator and \(Q(x)\) is the denominator. These components determine the behavior and properties of the rational function.

• The numerator, \(P(x)\), influences the zeros of the rational function. When \(P(x) = 0\), the rational function will be zero, provided the denominator is not zero at that point.
• The denominator, \(Q(x)\), plays a crucial role in the definition of the function, particularly for determining points of undefined values. If \(Q(x) = 0\), the rational function at that specific \(x\)-value is undefined, leading to vertical asymptotes or holes in its graph.

Mastery of these concepts is essential for the proper evaluation and interpretation of rational functions.

Zeros of Functions

Understanding the zeros of functions is essential in identifying where a function crosses the x-axis in a graph. For a rational function \(f(x) = \frac{P(x)}{Q(x)}\), the zeros are determined by solving \(P(x) = 0\) while ensuring that \(Q(x) eq 0\).

• Zeros of the Numerator: If \(x = c\) is a zero of the numerator \(P(x)\), it means that \(P(c) = 0\). The function value \(f(c)\) will equal zero only if \(Q(c) eq 0\). Otherwise, if \(Q(c) = 0\), \(f(c)\) would be undefined, as division by zero is impossible.
• Zeros of the Denominator: When \(Q(x) = 0\), the rational function is not defined at that value of \(x\). This leads to vertical asymptotes or holes, thus impacting the continuity of the graph.

Properly analyzing zeros provides insight into the behavior of the function, including where it may be undefined or where it intersects with the x-axis.