How to Simplify Rational Expressions Effectively

TL;DR

To simplify rational expressions, factor both the numerator and denominator, and then cancel any common factors. For operations such as addition or subtraction, identify the least common denominator and rewrite each fraction accordingly. Multiplication and division involve multiplying numerators and denominators directly or rewriting division as multiplying by the reciprocal.

Transcript

This video is about working with rational expressions. A rational expression is a fraction, usually with variables in it, something like x plus two over x squared minus three is a rational expression. In this video, we'll practice adding, subtracting, multiplying and dividing rational expressions, and simplifying them to lowest terms. We'll start w... Read More

Key Insights

• 📊 A rational expression is a fraction with variables in it, such as x plus two over x squared minus three.
• 📈 To simplify a rational expression to lowest terms, factor the numerator and denominator, then cancel out common factors.
• ✖️ When multiplying two fractions with just numbers, multiply the numerators and multiply the denominators.
• ➗ To divide two fractions, rewrite it as multiplying by the reciprocal of the fraction on the denominator.
• 👋 Instead of multiplying out the numerator and denominator, leave it in factored form to simplify and cancel common factors.
• 🔢 The least common denominator is the smallest expression that both denominators divide into.
• 📐 When finding the sum of two rational expressions, find the least common denominator and add the numerators.
• 🔄 One sided limits (limits from the left or from the right) can exist and be finite numbers or infinity (positive or negative).

Questions & Answers

Q: What is the difference between horizontal and vertical asymptotes in the graph of a rational function?

Horizontal asymptotes represent the end behavior of the function as x approaches positive or negative infinity and determine the horizontal line the graph approaches. Vertical asymptotes occur where the denominator of the rational function is equal to zero and represent the values of x where the graph approaches infinity or negative infinity.

Q: How are rational functions simplified to lowest terms?

To simplify a rational function to lowest terms, factor the numerator and denominator completely, and then cancel out any common factors. This helps to reduce the expression and remove any holes in the graph.

Q: How are horizontal asymptotes determined in a rational function?

Horizontal asymptotes in a rational function are determined by looking at the end behavior of the function as x approaches positive or negative infinity. The degree of the highest terms in the numerator and denominator helps determine the horizontal line the graph approaches.

Q: What are the key features of a rational function graph?

Key features of a rational function graph include the presence of horizontal and vertical asymptotes, holes in the graph where the expression is undefined, and the end behavior of the graph as x approaches positive or negative infinity.

Q: How do vertical asymptotes in a rational function affect the graph?

Vertical asymptotes occur where the denominator of the rational function is equal to zero. They represent the values of x where the function approaches infinity or negative infinity. The graph of the function gets infinitely close to the vertical line but never intersects it.

Summary & Key Takeaways

• A rational expression is a fraction with variables, and they can be simplified by factoring the numerator and denominator and canceling common factors.

• When graphing a rational function, the end behavior is determined by the terms with the highest exponents in the numerator and denominator.

• Horizontal asymptotes occur at the end behavior of the function, while vertical asymptotes occur at the values of x where the denominator is zero.

• Holes in the graph occur when the numerator and denominator have common factors that cancel out at certain x values.