Algebra and Functions – Rational Expressions: Simplifying – Simplifying algebraic fractions
Algebra and Functions – Rational Expressions: Simplifying – Simplifying Algebraic Fractions
Key Concepts
- Algebraic fractions, also known as rational expressions, are fractions in which the numerator and the denominator are both polynomials.
- These fractions can often be simplified by factoring and cancelling common factors in the numerator and the denominator.
- The process of simplifying an algebraic fraction is similar to that of simplifying a numeric fraction: you divide the numerator and the denominator by their highest common factor.
Factoring
- The first step in simplifying an algebraic fraction is typically to factor both the numerator and the denominator.
- Factoring breaks down the expressions into the product of their factors, which can simplify the process of identifying and cancelling common factors.
- Remember that a difference of squares, such as a^2 - b^2, can be factored into (a - b)(a + b).
Cancelling Common Factors
- Once the numerator and denominator have been factored, the next step is to identify and cancel any common factors.
- Common factors are expressions that appear in both the numerator and the denominator. By definition, any expression divided by itself is 1, so common factors can be cancelled out.
- This step is crucial: cancelling common factors can greatly simplify the fraction and highlight its important features.
Multiplying and Dividing Algebraic Fractions
- When multiplying algebraic fractions, you simply multiply the numerators together and the denominators together. Simplify if possible.
- When dividing algebraic fractions, you multiply by the reciprocal of the divisor. Realise that this is the same as multiplying by the fraction upside-down. Simplify if possible.
Adding and Subtracting Algebraic Fractions
- Like numeric fractions, algebraic fractions can only be added or subtracted if they have the same denominator (i.e., they are like fractions).
- If the fractions have different denominators, you will need to find a common denominator before you can add or subtract them.
Useful Strategies
- Practice makes perfect. Spend time working on a variety of problems involving algebraic fractions to solidify your understanding.
- Watch out for complex fractions - fractions where the numerator and/or denominator itself contains fractions. You'll need to simplify these before you can proceed.
- Stick with it. Simplifying algebraic fractions can be a complex process, but it's also a foundational skill in much of algebra. Mastering this will set you up for success in more complex topics.