How To Find The Zeros Of A Polynomial Fraction: A Complete Guide
Polynomial fractions are an integral part of higher mathematics, often used in various fields including engineering, physics, and finance. A polynomial fraction is a mathematical expression that contains a polynomial in the numerator and denominator. The zeros of a polynomial fraction are the points where the polynomial fraction is equal to zero.
Finding the zeros of a polynomial fraction can be a challenging task, and it requires a good understanding of the underlying concepts. In this article, we will discuss how to find the zeros of a polynomial fraction step by step. We will also cover some frequently asked questions about polynomial fractions.
Understanding Polynomial Fractions
Before we delve into how to find the zeros of a polynomial fraction, let’s review the concept of polynomial fractions. A polynomial fraction is a fraction where the numerator and denominator are polynomials. A polynomial is an algebraic expression made up of variables and coefficients, with no division by a variable.
For example, consider the polynomial fraction:
(3x^2 + 2x – 1)/(x^2 – 5)
Here, the numerator is the polynomial 3x^2 + 2x – 1, and the denominator is the polynomial x^2 – 5.
Steps To Find The Zeros Of A Polynomial Fraction
The first step in finding the zeros of a polynomial fraction is to factor the numerator and denominator. In some cases, you may have to use long division to factor the denominator. Once you have factored the numerator and denominator, you can cancel out any common factors between them.
Let’s take an example:
Find the zeros of the polynomial fraction:
(2x – 3)(x + 4)/(x – 2)
Step 1: Factorize the numerator and denominator.
(2x – 3)(x + 4)/(x – 2) = (2x – 3)(x + 4)/((x – 2))
Step 2: Cancel out any common factors in the numerator and denominator.
There are no common factors between the numerator and denominator.
Step 3: Find the zeros of the polynomial fraction.
To find the zeros of the polynomial fraction, you need to set the numerator equal to zero.
(2x – 3)(x + 4) = 0
This equation can be solved by the zero product property which states that if a and b are two real numbers, then if ab = 0, then either a = 0 or b = 0 or both.
Therefore,
2x – 3 = 0 or x + 4 = 0
2x = 3 or x = -4
x = 3/2 or x = -4.
Therefore, the zeros of the polynomial fraction (2x – 3)(x + 4)/(x – 2) are x = 3/2 and x = -4.
Frequently Asked Questions
Q: What is the difference between polynomial and polynomial fraction?
A: A polynomial is an algebraic expression made up of variables and coefficients, with no division by a variable. A polynomial fraction is a fraction where both the numerator and denominator are polynomials.
Q: How do you know if a polynomial fraction has a zero?
A: A polynomial fraction has a zero if the numerator of the fraction is equal to zero for some value of the variable.
Q: What is the zero product property?
A: The zero product property states that if a and b are two real numbers, then if ab = 0, then either a = 0 or b = 0 or both.
Q: Can a polynomial fraction have complex zeros?
A: Yes, a polynomial fraction can have complex zeros.
Conclusion
Finding the zeros of a polynomial fraction is an essential part of higher mathematics. In this article, we discussed how to find the zeros of a polynomial fraction by breaking down the process into three simple steps. By factoring the numerator and denominator and then canceling out any common factors, you can find the zeros of the polynomial fraction. We also covered some frequently asked questions about polynomial fractions, including the difference between a polynomial and a polynomial fraction, the zero product property, and whether a polynomial fraction can have complex zeros.