Page 2: Operations on Algebraic Fractions

This page delves into the various operations on algebraic fractions, including addition, subtraction, multiplication, and division.

Adding and subtracting algebraic fractions requires finding a common denominator. Once a common denominator is established, the numerators can be combined.

Example: $-3x + 1$ / 8 + x / 4 = $-3x + 1$ / 8 + (2x) / 8 = $-x + 1$ / 8

Multiplying algebraic fractions is straightforward – multiply the numerators and denominators separately.

Example: $3 / (x + 1)$ * $4 / x$ = (12) / $x^2 + x$

Dividing algebraic fractions involves multiplying by the reciprocal of the divisor.

Example: $4 / x$ ÷ $2 / 5$ = $4 / x$ * $5 / 2$ = (10) / (2x)

Highlight: When performing działania na ułamkach algebraicznych (operations on algebraic fractions), always be mindful of the domains and simplify the results when possible.

Page 3: Rational Equations

This page focuses on rational equations, which are equations involving algebraic fractions.

Solving rational equations involves three key steps:

1. Determine the domain of the equation
2. Solve the equation algebraically
3. Check if the solution is within the domain

Definition: A rational equation is an equation where at least one of the terms is a rational expression (an algebraic fraction).

When solving rational equations, it's crucial to find the common denominator and multiply both sides by it to eliminate fractions.

Example: For the equation $4x^2 + 9$ / $3x + 6$ = 0, multiply both sides by $3x + 6$ to get 4x^2 + 9 = 0

Highlight: Always check the domain of rational equations to avoid extraneous solutions. The domain excludes values that make any denominator zero.

Vocabulary: Równania wymierne (rational equations) require careful consideration of domains and often involve algebraic manipulation to solve.

Page 1: Algebraic Fractions and Operations

This page introduces the fundamental concepts of algebraic fractions and basic operations performed on them.

Expanding algebraic fractions involves multiplying both numerator and denominator by the same factor. For example, expanding 3x/4 to 16x/16 requires multiplying both top and bottom by 4.

Example: 3x/4 = $3x * 4$ / $4 * 4$ = 12x/16

Reducing algebraic fractions is the process of simplifying fractions by dividing both numerator and denominator by their common factors.

Example: $x^2 - 4$ / $3x + 6$ can be reduced to x + 2$$x - 2 / 3$x + 2$, which further simplifies to $x - 2$ / 3

The domain of algebraic fractions is crucial to consider, as it excludes values that make the denominator zero.

Highlight: Always determine the domain of algebraic fractions by setting the denominator not equal to zero and solving for the excluded values.

Simplification of complex fractions often involves factoring and canceling common terms between numerator and denominator.

Vocabulary: Skracanie ułamków algebraicznych (reducing algebraic fractions) is a key skill in working with these expressions.