Questions and Answers
What is the factored form of the expression $\frac{3x^2 - 14x + 8}{x^2 - 16}$?
What is the method to multiply algebraic fractions?
Multiply the numerators and denominators, then reduce to lowest terms.
In the expression $\frac{4x^2}{5y} \times \frac{10x}{y^2} \times \frac{y}{8x^2}$, what terms can be cross-cancelled?
4x^2 cancels with 8x^2, 10 cancels with 5, and y cancels with y.
What do we get after canceling like terms in $\frac{4x^2}{5y} \times \frac{10x}{y^2} \times \frac{y}{8x^2}$?
Signup and view all the answers
What is the factored form of the expression $\frac{-4x + 8}{3x + 6} \times \frac{2x + 4}{4x - 12}$?
Signup and view all the answers
What is the process called while dividing algebraic expressions?
Signup and view all the answers
What is the first step in adding/subtracting algebraic expressions?
Signup and view all the answers
The LCD of the fractions $\frac{1}{x^2 - x}$, $\frac{1}{x^2 - 1}$, and $\frac{1}{x^2}$ is $\frac{x^2(x - 1)(x + 1)}{__}$
Signup and view all the answers
What is the final expression after factoring $y^4 + 2y - 35$?
Signup and view all the answers
What is the LCD of the expression $\frac{3x}{a^2} + \frac{4}{ax}$?
Signup and view all the answers
Study Notes
Algebraic Fractions Overview
- Algebraic fractions consist of a numerator and denominator, which can be polynomials.
- Key operations include addition, subtraction, multiplication, division, and finding least common denominators (LCD).
Factoring Expressions
- To simplify fractions, factor both the numerator and denominator.
- Example: Simplifying ( \frac{3x^2 - 14x + 8}{x^2 - 16} ) results in ( \frac{(3x - 2)(x - 4)}{(x - 4)(x + 4)} ), which simplifies to ( \frac{3x - 2}{x + 4} ) after canceling ( (x - 4) ).
Multiplying Algebraic Fractions
- Multiply fractions by multiplying the numerators together and the denominators together.
- This can also involve canceling common factors before performing the multiplication.
Cross Cancellation
- When multiplying multiple fractions, like terms can be canceled:
- Example: In ( \frac{4x^2}{5y} \times \frac{10x}{y^2} \times \frac{y}{8x^2} ), canceling results in ( \frac{x}{y^2} ).
Dividing Algebraic Expressions
- Division of fractions involves inverting the second fraction and then multiplying, which facilitates simplification.
Adding/Subtracting Expressions
- Finding the LCD is crucial for adding or subtracting algebraic fractions.
- The LCD is determined by factoring each denominator and taking the highest power of each factor.
Steps to Find the LCD
- Factor all denominators thoroughly.
- The LCD is the product of distinct factors taken at their maximum occurrence found in any denominator.
Example LCD Calculation
- For ( \frac{1}{x^2 - x} ), ( \frac{1}{x^2 - 1} ), and ( \frac{1}{x^2} ):
- Factored forms yield ( x(x - 1) ), ( (x + 1)(x - 1) ), and ( x^2 ).
- The final LCD is ( x^2(x - 1)(x + 1) ).
Process for Adding/Subtracting Algebraic Fractions
- Step 1: Identify the LCD.
- Step 2: Rewrite fractions over the LCD.
- Step 3: Combine numerators over the LCD.
- Step 4: Simplify the resulting expression if possible.
Example of Simplification
- For ( \frac{y-3}{y-5} + \frac{y - 23}{y^2 - y - 20} ), factor and combine:
- LCD found as ( (y-5)(y+4) ).
- Final expression after simplification leads to ( \frac{y + 7}{y + 4} ) with the condition that ( y \neq 5 ).
General Tips
- Ensure thorough factorization before performing operations to identify possible cancellations.
- Always check for undefined values (e.g., denominators equating to zero) before simplifying fractions.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz focuses on algebraic fractions, including factoring expressions and multiplying fractions. Study how to simplify fractions and reduce to lowest terms through detailed flashcards. Perfect for revising essential algebraic operations.
