This lesson builds on your knowledge of numerical fractions and basic algebra. The focus is on multiplying and dividing fractions that contain algebraic terms. The single most important skill in this topic is factorizing.

1. Core Principle: Factorize First!

Before you multiply or divide, always factorize every numerator and denominator that can be factorized. This includes:

• Taking out common factors (e.g., 2x + 4 becomes 2(x + 2)).

• Difference of two squares (e.g., x² - 9 becomes (x - 3)(x + 3)).

• Trinomial quadratic expressions (e.g., x² - 5x + 6 becomes (x - 2)(x - 3)).

Why? Factorizing reveals the common terms (like (x+2)) that you can cancel out, which makes the problem much simpler.

2. Multiplying Algebraic Fractions

The rule is the same as for regular fractions: (Top × Top) / (Bottom × Bottom).

Steps to Solve:

1. Factorize all numerators and denominators completely.

2. Cancel any common factors that appear in both a numerator and a denominator.

3. Multiply the remaining terms in the numerators together.

4. Multiply the remaining terms in the denominators together.

Example: Simplify (a² - 9) / 5a * (2a - 4) / (a² + a - 6)

1. Factorize: = [(a - 3)(a + 3)] / 5a * [2(a - 2)] / [(a + 3)(a - 2)]

2. Cancel: The (a + 3) and (a - 2) terms appear on both the top and bottom, so they cancel out. = [(a - 3)<s>(a + 3)</s>] / 5a * [2<s>(a - 2)</s>] / [<s>(a + 3)</s><s>(a - 2)</s>]

3. Multiply Remainder: = (a - 3) * 2 / 5a = 2(a - 3) / 5a

3. Dividing Algebraic Fractions

The rule is the same as for regular fractions: "Invert (flip) the second fraction and multiply."

Steps to Solve:

1. Change the division sign (÷) to a multiplication sign (×).

2. Flip the second fraction (the one after the sign). This is called finding its reciprocal.

3. Now the problem is a multiplication problem. Follow the steps for multiplication: Factorize, Cancel, Multiply.

Example: Simplify (x² + 3x - 10) / x ÷ (x² - 25) / (x² - 5x)

1. Invert and Multiply: = (x² + 3x - 10) / x * (x² - 5x) / (x² - 25)

2. Factorize: = [(x + 5)(x - 2)] / x * [x(x - 5)] / [(x - 5)(x + 5)]

3. Cancel: Cancel the (x + 5), (x - 5), and x terms. = [<s>(x + 5)</s>(x - 2)] / <s>x</s> * [<s>x</s><s>(x - 5)</s>] / [<s>(x - 5)</s><s>(x + 5)</s>]

4. Result: All that's left is (x - 2). = x - 2

Exam Tips & Common Mistakes

• Mistake 1: Multiplying Before Factoring. If you multiply large expressions together first, you will get a very complicated fraction that is almost impossible to simplify. ALWAYS factorize first.

• Mistake 2: Incorrect Cancelling. You can only cancel a factor if it's identical in the numerator and the denominator. You cannot cancel the x² from (x² + 1) and x². You can only cancel the entire bracket (x² + 1).

• Mistake 3: Forgetting to Flip. When dividing, students sometimes forget to invert the second fraction. Write it down as your very first step to avoid this.

• Exam Tip: Be an expert at factorizing. This is the most important skill for this lesson. If you are slow or make mistakes in factorizing, you will not be able to solve these problems. Revise the difference of two squares and trinomials.