This is a very important foundational lesson. Mastering the skill of finding the LCM of algebraic expressions is essential for adding and subtracting the algebraic fractions you will see in the next lesson.

1. Core Concepts (Short Notes)

• LCM of Algebraic Expressions: The smallest algebraic expression that is perfectly divisible by each of the given expressions.

• The Golden Rule: The LCM is the product of the highest power of every distinct factor present in the expressions.

2. Key Methods

Finding the LCM is a two-part process: first handle the numbers (coefficients), then the variables and brackets.

Method 1: For Simple Algebraic Terms (e.g., 4a², 6ab, 8b)

1. Factorize Each Term: Break down each term into its prime factors and variables.

   • 4a² = 2² × a²

   • 6ab = 2 × 3 × a × b

   • 8b = 2³ × b

2. Find the LCM of the Coefficients: Look at the prime factors of the numbers (2 and 3). Take the highest power of each.

   • Highest power of 2 is 2³ (from 8b).

   • Highest power of 3 is 3¹ (from 6ab).

   • LCM of numbers = 2³ × 3 = 8 × 3 = 24.

3. Find the LCM of the Variables: Look at the variables (a and b). Take the highest power of each.

   • Highest power of a is a² (from 4a²).

   • Highest power of b is b¹ (from 6ab and 8b).

   • LCM of variables = a²b.

4. Combine Them: Multiply the results from steps 2 and 3.

   • Final LCM = 24a²b

Method 2: For Expressions with Brackets (e.g., 2x-6, x²-9)

1. FACTORIZE EACH EXPRESSION FIRST! This is the most important step. You cannot find the LCM until everything is in its simplest factored form.

   • 2x - 6 = 2(x - 3)

   • x² - 9 = (x - 3)(x + 3) (This is the difference of two squares!)

2. Identify All Distinct Factors: List every unique factor you see.

   • The distinct factors are: 2, (x - 3), and (x + 3).

3. Take the Highest Power of Each Factor: In this case, all factors have a power of 1.

4. Combine Them: Multiply the factors together.

   • Final LCM = 2(x - 3)(x + 3)

3. Tips & Tricks for Exams

• Always Factorize First: This is the number one rule. If you see expressions like 3x+6 or x²-4, your first instinct must be to factorize them to 3(x+2) and (x-2)(x+2).

• Spot the Difference of Two Squares: a² - b² = (a - b)(a + b). This is a very common pattern in exam questions (e.g., x²-25, 4a²-9).

• Handle (a-b) and (b-a): Remember that (b - a) = -1(a - b). This isn't usually needed for finding the LCM, but it's critical for the next lesson on fractions.

• Highest Power Wins: When you have factors like (x-1) and (x-1)², the LCM must include the highest power, which is (x-1)². The smaller power is ignored.

  • Example: LCM of 3(x-1) and 5(x-1)² is 15(x-1)².

4. Important Points to Remember

• Do not confuse LCM with HCF (Highest Common Factor). The HCF only uses the lowest powers of common factors. The LCM uses the highest powers of all factors.

• The brackets, like (x+2), are treated as single, indivisible factors. You cannot take parts from inside a bracket.

• Practice this well. If you cannot find the LCM correctly, you will not be able to solve the problems in the next lesson.