Rational Expressions and Fractions

• A fraction can also be referred to as a ratio.

• Example: \frac{15}{65}.

Simplifying Rational Expressions

• Simplification process involves factoring the numerator and denominator first and then canceling out common factors.

• Example:

• Expression: \frac{(x + 2)^2}{x^2 - 4}

• Factor numerator: x + 2 imes x + 2

• Factor denominator using difference of squares: x^2 - 4 = (x + 2)(x - 2)

• Cancel out common terms: \frac{(x + 2)\cancel{(x + 2)}}{\cancel{(x + 2)}(x - 2)} = \frac{x + 2}{x - 2}

Example 1 Insights

• Expression for a: \frac{2x^2}{10x^3 - 2x^2}

• Numerator cannot be factored; bring 2x^2 over.

• Denominator: Factor out the greatest common factor (GCF).

• Result: 2x^2\cdot (5x - 1)

Factoring Strategy

• For b: Simplify using factoring and GCF, e.g., 9*x^2 can be factored into terms with 36 (combinations of factors).

• Key point is the final expression:

• Determine x values from factoring.

Commutative Property

• In addition, the order of terms does not matter (commutative property).

• Example:

• Terms: 2 + x can be rewritten as x + 2 for cancellation.

Negative Signs in Expressions

• When expressions look different, but share common elements with a negative sign, factor out negative to simplify.

• Example a:

  • -(x - 2 + 2) becomes -x and will help in simplification.

Expressions with Perfect Squares

• Recognizing perfect squares (like 9 - x^2) allows for further factoring

• Resultant factors ger into a difference of squares.

Multiplying and Dividing Rational Expressions

• For multiplication, multiply across the numerator and denominator (straight multiplication).

• Example of complex rational expressions with necessary cancellations.

• When dividing, flip the second fraction and proceed similar to multiplication.

Least Common Denominator (LCD)

• Must find LCD before adding/subtracting fractions with different denominators.

• Example: Find LCD for \frac{x}{4} and \frac{x + 1}{3y^2}:

• For a single term denominator, simply multiply terms for the LCD.

Examples from Lecture

• Finding individual factors and LCD using binomials (e.g., m^2 - 25 = (m + 5)(m - 5)) to prepare for additions and computations across expressions.