Purchase Solution

Function of polynomial ratios

Not what you're looking for?

Ask Custom Question

Now consider a rational function, which is the ratio of two polynomials. These two polynomials will each have a set of zeros, and note that at a zero of the denominator we are actually dividing by zero. The zeros of the denominator are called poles, and they are point where the rational function becomes infinite (unless there is also a zero of the numerator at that point, which is called a zero of the function).
Describe the behavior of a rational function near one of its poles on the real axis. How does the function vary as the pole is approached from each side, and what happens at the pole? If there is also a zero at this point, then what happens at the pole? We know that zero divided by zero is indeterminate, but if there is a zero and a pole at the same point, what is a sensible definition of the function at this point?

Purchase this Solution
Solution Summary

This provides an example of giving a definition for a function that is the ratio of polynomials.

Solution Preview

A rational function of variable x can be written as p(x)/q(x) where p and q are both polynomial functions of x.

Describe the behavior of a rational function near one of its poles on the real axis?
As x approaches a zero of q(x), the actual value of q(x) becomes very small. Since q(x) is a continuous function, we can approach as close to the zero as we want and hence q(x) can become arbitrarily small in magnitude. Hence, p(x)/q(x) will approach + infinity or -infinity as this happens depending on the sign of p(x) on the side from which we ...

Purchase this Solution
Free BrainMass Quizzes
Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Probability Quiz

Some questions on probability

View More Free Quizzes
Related BrainMass Solutions

  • Rational functions, polynomials

    253851 Rational Functions and Polynomials Explain what makes a function a polynomial. Give an example of a function that is a polynomial and a function that is not a polynomial.

  • Analyzing a Polynomial Function

    Is the leading coefficient of the polynomial positive or negative? Why? D. What is the y-intercept of P? E. Construct a polynomial function whose zeros are those determined in Part A and whose y-intercept is the one determined in Part D. F.

  • Properties of Continuous Functions

    362895 Continuous functions 1.Show that a polynomial function is continuous at every point of R. 2. Show that a rational function (a quotient of two polynomial functions) at every point where it is defined. Please see the attachment. 1.

  • Finding Polynomials From Complex Roots

    165056 Finding Polynomials From Complex Roots Write a polynomial function of minimum degree in standard form with real coeffiecients whose zeros include those listed. 6. 1-2i and 1+2i 12. -2 and 1+2i Write a polynomial function of minimum degree

  • Real Analysis: Proof by induction that a polynomial of degree n>0 has at most n roots.

    9669 Proof by induction that a polynomial of degree n>0 has at most n roots. A polynomial of degree n>0 has at most n roots. (A root of a function is a point at which the function has value 0.) I need a proof by induction to show this.

  • Chebyshev polynomial interpolation

    Construct the degree four Chebyshev polynomial. Evaluate the resulting polynomial at x=0.8 and compare with the actual value of the function. Please see the attached file.

  • Writing a Matlab Function for Cubic Polynomial Interpolation Algorithm

    Note that when 's and 's are fixed, is a function of x. Plot this function. 3. Compare the function in (2) with the cubic polynomial interpolating . ( should be the value of the cubic interpolating polynomial at x.)

  • difference between a polynomial function and an exponential function

    What is the fundamental difference between a polynomial function and an exponential function? Be sure to include a description of the graph of each type of function. Please see attached.

  • Metric, Measure and Polynomial

    If , show that there is a measure on [0,1] such that for every polynomial of degree at most , .(1) Solution: Define a polynomial function: Then we write Define a measure :[0,1]  R with {[0,1]} = 1 Two ways

  • Modeling data with polynomial functions and rational functions

    The solution includes definitions of the terms "function", "polynomial function", and "rational function". It explains through examples how to use Excel to model data using polynomial and rational functions.

View More Related Solutions