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Chapter 6: Problem 18

A fraction that has polynomials in its numerator and denominator is called a _____ expression.

Short Answer

Expert verified
Rational

Step by step solution

01

Understanding the Given Information

Here, a fraction is given where both the numerator and the denominator are polynomials. The task is to find the name of such an expression.
02

Naming the Expression

The fraction with polynomials in both the numerator and the denominator is called a Rational expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomials
Polynomials are mathematical expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication operations. They do not include division by a variable.
  • A polynomial looks like this: \( x^2 + 4x + 4 \).
  • Each component of the polynomial is referred to as a "term."
  • The degree of a polynomial is the highest power of the variable, for example, the degree of \( x^2 + 4x + 4 \) is 2.
Polynomials can have multiple terms:
  • Monomial 鈥� contains one term (e.g., \( 3x \)).
  • Binomial 鈥� contains two terms (e.g., \( x^2 + 3 \)).
  • Trinomial 鈥� contains three terms (e.g., \( x^2 + 4x + 4 \)).
Polynomials form the basis of more complex algebraic operations and have many practical applications in fields such as physics and engineering.
Numerator
The numerator is the top part of a fraction. In the case of rational expressions, the numerator is a polynomial.
  • For instance, in the expression \( \frac{2x + 1}{x - 3} \), \( 2x + 1 \) is the numerator.
  • The numerator indicates how many parts of the whole are being considered.
When dealing with rational expressions, it's important to simplify the numerator as much as possible:
  • Factor the polynomial if applicable.
  • Cancel out any common factors with the denominator.
Understanding the numerator's structure and content can significantly aid in solving rational expressions more effectively.
Denominator
The denominator is the bottom part of a fraction, and in rational expressions, it is a polynomial. It states the total number of equal parts the whole is divided into.
  • In the expression \( \frac{2x + 1}{x - 3} \), \( x - 3 \) is the denominator.
Working with the denominator involves ensuring it is not equal to zero:
  • If the denominator is zero, the expression is undefined as division by zero is not possible.
  • It's common practice to state the restrictions of a rational expression by noting the values for which the denominator equals zero.
Similar to the numerator, simplify the denominator when possible by factoring. This simplification can expose common factors with the numerator, making it easier to reduce the rational expression to its simplest form.

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