key term - Factor

Definition

A factor is a number or expression that divides another number or expression without a remainder. Factors are fundamental building blocks that can be used to break down and understand more complex mathematical expressions, particularly in the context of rational functions.

5 Must Know Facts For Your Next Test

1. Factors are essential in simplifying rational functions by breaking down the numerator and denominator into their prime factors.
2. The factors of a rational function's numerator and denominator determine the behavior of the function, such as its domain, asymptotes, and end behavior.
3. Factoring a rational function's numerator and denominator can reveal important information about the function, such as its zeros and points of discontinuity.
4. Identifying common factors between the numerator and denominator can help simplify a rational function by canceling out shared factors.
5. The degree of a rational function is determined by the highest exponent of the factors in both the numerator and denominator.

Review Questions

• Explain how factors are used to simplify rational functions.
  • Factors are crucial in simplifying rational functions because they allow you to break down the numerator and denominator into their prime components. By identifying common factors between the numerator and denominator, you can cancel them out, which reduces the complexity of the rational function and can reveal important information about its behavior, such as its domain, asymptotes, and end behavior.
• Describe the relationship between the factors of a rational function and its degree.
  • The degree of a rational function is determined by the highest exponent of the factors in both the numerator and denominator. The numerator and denominator can be factored to identify the individual factors and their corresponding exponents. The degree of the rational function is then calculated as the difference between the highest exponent in the numerator and the highest exponent in the denominator. This relationship between factors and degree is crucial for understanding the overall behavior and properties of a rational function.
• Analyze how the factorization of a rational function's numerator and denominator can provide insights into the function's critical points and discontinuities.
  • $$The factorization of a rational function's numerator and denominator can reveal important information about the function's critical points and discontinuities. By identifying the factors, you can determine the zeros of the function, which correspond to the x-intercepts or points where the function is equal to zero. Additionally, the factors can indicate points of discontinuity, which occur when the denominator is equal to zero. Understanding the factorization of the numerator and denominator is essential for analyzing the behavior and properties of a rational function, as it allows you to identify its critical points, asymptotes, and other important characteristics.$$

Related terms

Prime Factor: A prime factor is a factor that is a prime number, meaning it is only divisible by 1 and itself.

Factorization:

Factorization is the process of expressing a number or expression as a product of its factors.

Greatest Common Factor (GCF):

The greatest common factor is the largest factor that is shared by two or more numbers or expressions.