Pre-Calculus 11 Notes

By: Gen L.

Lesson 1: Rational Expressions Basics

What is a Rational Expression? Part 1

• A Rational number is any number that can be expressed as a ratio () of two integers: and , where ( is non-zero).
  • Example:
• An expression is a mathematical phrase containing numbers, variables and operators (polynomials)
  • Example: .

What is a Rational Expression? Part 2

• A Rational Expression is nothing more than a ratio where and are polynomials
• They must also follow the same rules which govern fractions.
  • LCD ()
  • Answer in lowest terms (Simplify)
  • Cross Cancel ()
  • Kiss & Flip ()

Simplifying Rational Expressions, Part 1

• Ex.

• When we simplify, we break the numerator and denominator into base components and determining what "cancels out". This works for both basic fractions & rational expressions.

Simplifying Rational Expressions, Part 2

• Simplifying works with more complicated scenarios as well:
• Even when dealing with larger terms (or brackets), same rules apply.
• We may need to modify our questions however.

Simplifying Rational Expressions, Part 3

• Take this example: . Here's how to solve:

1. Factor
2. Cancel
3. Clean

Edge Values

• Look at the graph of .
• There's something strange at , there's an invisible line where the graph is undefined.
• This is called a Vertical Asymptote. Why is it where it is?
• If we substitute into , we get .
• Since is undefined, we say it's a Non-Permissable Value (NPV).
• When dealing with Rationals, we have to check for these NPVs, as they do not exist.

Toeing the Edge: Finding NPVs

• To find NPVs, we only need to check the denominators.

• Examples:

• When looking at NPVs, they can appear at ALL steps of solving. If an NPV is possible, we MUST report them.

Example A: Rationals & NPVs

• State the NPV(s) and Simplify:

1. Factor
2. State NPVs

Example A, Cont.

3. Simplify:

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