Simplify rational expressions

Learning Objectives Identify a rational expression and its components. Determine the non-permissible values (NPVs) for a given rational expression. Factor complex polynomials in the numerator and denominator, including trinomials, difference of squares, and sum/difference of cubes. Correctly identify and cancel common factors between the numerator and denominator. Write a rational expression in its simplest form. Recognize when a rational expression is already in its simplest form. Ever felt like you're looking at an algebraic monster? 🐲 Simplifying rational expressions is like finding the secret spell to reveal its much simpler, true form! This tutorial will guide you through the process of simplifying rational expressions, which are essentially fractions containing...

TermDefinitionExample Rational ExpressionA fraction in which the numerator and the denominator are polynomials. The denominator cannot be equal to zero.(x^2 + 5x + 6) / (x + 3) NumeratorThe polynomial in the top part of the rational expression.In (x - 2) / (x + 5), the numerator is (x - 2). DenominatorThe polynomial in the bottom part of the rational expression.In (x - 2) / (x + 5), the denominator is (x + 5). Non-Permissible Values (NPVs)Values of the variable that would make the denominator of a rational expression equal to zero. These values are excluded from the domain.For the expression (x - 2) / (x + 5), the NPV is x = -5, because it would make the denominator zero. FactorA polynomial that divides another polynomial evenly. We break down complex polynomials into a product of simpler...

Fundamental Principle of Rational Expressions \frac{P \cdot K}{Q \cdot K} = \frac{P}{Q}, \text{ where } Q \neq 0, K \neq 0 This is the core rule for simplification. If the numerator and denominator share a common factor (K), that factor can be cancelled out. This is only possible after the polynomials (P, Q, K) have been fully factored. Difference of Squares a^2 - b^2 = (a - b)(a + b) A frequently used factoring pattern. Recognizing this pattern is essential for quickly factoring binomials in the numerator or denominator. Factoring Trinomials ax^2 + bx + c = (mx + p)(nx + q) To simplify most rational expressions, you must be proficient at factoring trinomials into two binomials. This involves finding two numbers that multiply to 'ac' and add to 'b&#039...