Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Expand and simplify expressions involving binomials and trinomials.
• Factorise expressions using common factors, difference of two squares, and trinomials.
• Apply factorisation and simplification to solve problems.
• Recognize and factorise by grouping.

Lesson summary

This week, we delve deeper into algebraic expressions, building upon what we learned previously. Algebra is not just an abstract concept; it is a fundamental tool for problem-solving in various real-life situations, from calculating the cost of groceries to understanding interest rates on loans. Being proficient in manipulating algebraic expressions allows you to model and analyze real-world scenarios, making informed decisions. For example, understanding how to simplify expressions can help you optimize resource allocation in a small business or understand the impact of inflation on your savings.

Lesson notes

2.1 Expanding Algebraic Expressions Expanding algebraic expressions involves removing brackets by multiplying each term inside the bracket by the term outside. The distributive property is the core principle here: a(b + c) = ab + ac. When dealing with binomials and trinomials, you might need to apply the FOIL method (First, Outer, Inner, Last) or similar strategies to ensure you multiply each term correctly.

Distributive Property: As mentioned, a(b + c) = ab + ac. This extends to more complex cases like a(b + c + d) = ab + ac + ad. Also, remember the sign rules: positive x positive = positive, negative x negative = positive, positive x negative = negative, and negative x positive = negative.

Expanding Binomials (FOIL Method): When expanding (a + b)(c + d), FOIL stands for: First: a x c = ac Outer: a x d = ad Inner: b x c = bc Last: b x d = bd Therefore, (a + b)(c + d) = ac + ad + bc + bd Expanding Binomial Squares: (a + b)² = (a + b)(a + b) = a² + 2ab + b² and (a - b)² = (a - b)(a - b) = a² - 2ab + b² Expanding Trinomials: Requires careful application of the distributive property. For instance, (a + b)(c + d + e) = a(c + d + e) + b(c + d + e) = ac + ad + ae + bc + bd + be Example 1: Expand and simplify 2x(x - 3) + 4(x + 1)

Step 1: Apply the distributive property: 2x² - 6x + 4x + 4 Step 2: Combine like terms: 2x² - 2x + 4 Example 2: Expand and simplify (x + 2)(x - 3)

Step 1: Apply the FOIL method: x² - 3x + 2x - 6 Step 2: Combine like terms: x² - x - 6 Example 3: Expand and simplify (2a - 1)² Step 1: Write as (2a - 1)(2a - 1)

Step 2: Apply the FOIL method: 4a² - 2a - 2a + 1 Step 3: Combine like terms: 4a² - 4a + 1 2.2 Factorising Algebraic Expressions Factorising is the reverse of expanding. It involves expressing an algebraic expression as a product of its factors. Different techniques are used depending on the type of expression.

Common Factor: Identify the greatest common factor (GCF) of all terms and factor it out.

Example: Factorise 6x² + 9x. The GCF is 3x. So, 6x² + 9x = 3x(2x + 3)

Difference of Two Squares: a² - b² = (a + b)(a - b)

Example: Factorise x² -

1

6. This is a difference of two squares (x² and 4²). So, x² - 16 = (x + 4)(x - 4)

Trinomials: Trinomials of the form ax² + bx + c can be factorised into two binomials. The key is to find two numbers that multiply to give 'c' and add to give 'b' (if a = 1).

Example: Factorise x² + 5x +

6. We need two numbers that multiply to 6 and add to

5. These numbers are 2 and

3. So, x² + 5x + 6 = (x + 2)(x + 3) If a ≠ 1, the process is more complex and often involves trial and error or the "ac method".

Example: Factorise 2x² + 7x +

3. We need to find two numbers that multiply to 2*3 = 6 and add to

7. These numbers are 6 and

1. So, 2x² + 7x + 3 = 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)

Factorising by Grouping: This is used when there are four or more terms and no single common factor for all terms. Group terms in pairs and factor out a common factor from each pair. Then, factor out the common binomial factor.

Example: Factorise ax + ay + bx + by.

Step 1: Group the terms: (ax + ay) + (bx + by)

Step 2: Factor out common factors from each group: a(x + y) + b(x + y)

Step 3: Factor out the common binomial factor: (x + y)(a + b) 2.3 Simplifying Algebraic Expressions Simplifying involves combining like terms after expanding and/or factorising. This reduces the expression to its simplest form. Like terms are terms that have the same variable raised to the same power.

Example: Simplify 3x² + 2x - x² + 5x - 4 Step 1: Group like terms: (3x² - x²) + (2x + 5x) - 4 Step 2: Combine like terms: 2x² + 7x - 4 Guided Practice (With Solutions)

Question 1: Expand and simplify: 3(x - 2) + 2(x + 1)

Solution: Step 1: Apply the distributive property: 3x - 6 + 2x + 2 Step 2: Combine like terms: 5x - 4

Commentary: This question reinforces the basic distributive property and combining like terms. It highlights the importance of paying attention to signs.

Question 2: Factorise completely: 4x² - 9 Solution: Step 1: Recognize the difference of two squares: (2x)² - (3)² Step 2: Apply the formula a² - b² = (a + b)(a - b): (2x + 3)(2x - 3)

Commentary: This question tests the understanding of the difference of two squares factorization. Students should be able to identify the perfect squares and apply the formula correctly.

Question 3: Expand and simplify: (x + 3)(x - 4)

Solution: Step 1: Apply the FOIL method: x² - 4x + 3x - 12 Step 2: Combine like terms: x² - x - 12

Commentary: This question reinforces the FOIL method for expanding binomials and combining like terms. Pay close attention to the signs when multiplying.

Question 4: Factorise completely: x² - 8x + 15 Solution: Step 1: Find two numbers that multiply to 15 and add to -

8. These numbers are -3 and -

5. Step 2: Write the factorised form: (x - 3)(x - 5)

Commentary: This question tests the ability to factorise trinomials where a = 1.