Algebraic Fraction

Definition

An algebraic fraction is a mathematical expression that represents a ratio of two algebraic expressions, where the numerator and denominator are both polynomials. Algebraic fractions are a fundamental concept in algebra and are essential for understanding and manipulating rational expressions, which are the focus of topics such as simplifying rational expressions, simplifying complex rational expressions, and solving rational equations.

5 Must Know Facts For Your Next Test

1. Algebraic fractions can be simplified by factoring the numerator and denominator and then canceling common factors.
2. Complex rational expressions are fractions that contain other fractions in either the numerator or the denominator, or both.
3. Solving rational equations involves finding the values of the variable(s) that make the equation true, which may require techniques such as cross-multiplication or finding a common denominator.
4. The degree of an algebraic fraction is determined by the highest degree of the polynomial in the numerator and denominator.
5. Algebraic fractions can be used to represent and solve real-world problems involving ratios, rates, and proportions.

Review Questions

• Explain the process of simplifying an algebraic fraction.
  • To simplify an algebraic fraction, you need to factor the numerator and denominator to identify any common factors. Then, you can cancel out the common factors, leaving you with the simplified form of the fraction. This process reduces the fraction to its lowest terms and makes it easier to work with in further algebraic operations.
• Describe the key differences between simplifying a rational expression and simplifying a complex rational expression.
  • The main difference between simplifying a rational expression and a complex rational expression is the level of complexity. A rational expression is a fraction with a polynomial in the numerator and a polynomial in the denominator. Simplifying a rational expression involves factoring the numerator and denominator and canceling common factors. In contrast, a complex rational expression is a fraction that contains other fractions, either in the numerator, the denominator, or both. Simplifying a complex rational expression requires additional steps, such as finding a common denominator for the nested fractions and then performing the necessary algebraic operations to simplify the expression.
• Analyze the process of solving a rational equation and explain how it differs from solving a linear or polynomial equation.
  • Solving a rational equation involves finding the values of the variable(s) that make the equation true. This process differs from solving linear or polynomial equations because rational equations can have extraneous solutions, which are solutions that do not satisfy the original equation. To solve a rational equation, you may need to use techniques such as cross-multiplication or finding a common denominator, and then checking for extraneous solutions. The presence of the fraction(s) in the equation adds an extra layer of complexity compared to solving linear or polynomial equations, which do not involve fractions.

Related terms

Numerator: The numerator of an algebraic fraction is the polynomial expression that appears above the fraction bar, representing the dividend or the part being divided.

Denominator: The denominator of an algebraic fraction is the polynomial expression that appears below the fraction bar, representing the divisor or the part that is dividing.

Rational Expression: A rational expression is a quotient of two polynomial expressions, where the denominator is not allowed to be zero.