Algebraic fractions are mathematical expressions that represent a ratio of two algebraic expressions, typically consisting of a numerator and a denominator. These fractions are used to perform operations such as addition, subtraction, multiplication, and division within the context of algebra.
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Algebraic fractions can be simplified by factoring the numerator and denominator to find common factors that can be cancelled.
When adding or subtracting algebraic fractions with a common denominator, the numerators are added or subtracted, and the common denominator remains the same.
To add or subtract algebraic fractions with unlike denominators, the fractions must first be converted to equivalent fractions with a common denominator, typically the least common denominator (LCD).
Multiplying algebraic fractions involves multiplying the numerators and multiplying the denominators, then simplifying the resulting fraction.
Dividing algebraic fractions involves inverting the second fraction and then multiplying the first fraction by the inverted second fraction.
Review Questions
Explain the process of adding or subtracting algebraic fractions with a common denominator.
To add or subtract algebraic fractions with a common denominator, the numerators are simply added or subtracted, while the common denominator remains the same. For example, to add the fractions $\frac{2x}{3}$ and $\frac{5x}{3}$, the numerators are added: $\frac{2x}{3} + \frac{5x}{3} = \frac{7x}{3}$. This method works because the denominators are the same, allowing the fractions to be combined directly.
Describe the steps involved in adding or subtracting algebraic fractions with unlike denominators.
When adding or subtracting algebraic fractions with unlike denominators, the fractions must first be converted to equivalent fractions with a common denominator, typically the least common denominator (LCD). To do this, the numerator and denominator of each fraction are multiplied by the appropriate factor to obtain the LCD. Then, the numerators are added or subtracted, and the common denominator is retained. For example, to add $\frac{2x}{3}$ and $\frac{5y}{4}$, the LCD would be 12, and the fractions would become $\frac{8x}{12}$ and $\frac{15y}{12}$, which can then be added to obtain $\frac{23y}{12}$.
Analyze the process of multiplying and dividing algebraic fractions, and explain the importance of simplifying the resulting fractions.
When multiplying algebraic fractions, the numerators are multiplied, and the denominators are multiplied. The resulting fraction is then simplified by factoring the numerator and denominator to cancel any common factors. Dividing algebraic fractions involves inverting the second fraction and then multiplying the first fraction by the inverted second fraction. Simplifying the resulting fraction is crucial because it reduces the expression to its most basic form, making it easier to work with and interpret. Simplifying algebraic fractions can also reveal important properties or relationships within the expression, such as common factors or the presence of perfect squares.