What are the zeros of a rational function?
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Zoey Adams
Studied at Princeton University, Lives in Princeton, NJ
As a domain expert in mathematics, I can explain that the zeros of a rational function are the values of the independent variable (usually denoted as \( x \)) that make the function equal to zero. A rational function is a function that can be expressed as the quotient of two polynomial functions:
\[ R(x) = \frac{P(x)}{Q(x)} \]
where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not the zero polynomial (to avoid division by zero).
The zeros of the rational function \( R(x) \) are found by setting the numerator \( P(x) \) equal to zero and solving for \( x \), provided that \( Q(x) \) is not zero at those points (since division by zero is undefined):
\[ P(x) = 0 \quad \text{and} \quad Q(x) \neq 0 \]
These solutions are also known as the horizontal asymptotes if \( P(x) \) and \( Q(x) \) have the same degree, or the x-intercepts if the function crosses the x-axis at those points.
Important Note: The zeros of the rational function are not the same as the zeros of the denominator \( Q(x) \). The denominator cannot be zero because that would make the function undefined.
\[ R(x) = \frac{P(x)}{Q(x)} \]
where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not the zero polynomial (to avoid division by zero).
The zeros of the rational function \( R(x) \) are found by setting the numerator \( P(x) \) equal to zero and solving for \( x \), provided that \( Q(x) \) is not zero at those points (since division by zero is undefined):
\[ P(x) = 0 \quad \text{and} \quad Q(x) \neq 0 \]
These solutions are also known as the horizontal asymptotes if \( P(x) \) and \( Q(x) \) have the same degree, or the x-intercepts if the function crosses the x-axis at those points.
Important Note: The zeros of the rational function are not the same as the zeros of the denominator \( Q(x) \). The denominator cannot be zero because that would make the function undefined.
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Studied at the University of Amsterdam, Lives in Amsterdam, Netherlands.
When a rational function is equal to zero (that is, its output is equal to zero) then its NUMERATOR is equal to zero. So, to find the zeros of a rational function we simply find the zeros of the NUMERATOR. EXAMPLE: The zeros of the function h(x) described above would be found by setting the NUMERATOR equal to zero.
2023-04-08 08:03:58
Charlotte Perez
QuesHub.com delivers expert answers and knowledge to you.
When a rational function is equal to zero (that is, its output is equal to zero) then its NUMERATOR is equal to zero. So, to find the zeros of a rational function we simply find the zeros of the NUMERATOR. EXAMPLE: The zeros of the function h(x) described above would be found by setting the NUMERATOR equal to zero.
