Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Simplify algebraic expressions by combining like terms.
• Factorize by removing a common factor.
• Factorize a difference of two squares.
• Factorize trinomials of the form x² + bx + c.
• Apply these skills to solve problems.

Lesson summary

Algebraic expressions and factorization are fundamental building blocks in mathematics. They are not just abstract concepts; they are crucial tools for solving real-world problems in various fields, from engineering and finance to everyday budgeting and resource allocation. In a South African context, understanding algebraic expressions helps in areas like calculating profit margins in small businesses, determining optimal land usage for agriculture, and analyzing financial data for community development projects. Factorization, in particular, simplifies complex problems by breaking them down into manageable components.

Lesson notes

2.1 Algebraic Expressions: Building Blocks An algebraic expression is a combination of variables, constants, and mathematical operations (addition, subtraction, multiplication, division, exponents, etc.).

Variable: A symbol (usually a letter) representing an unknown quantity (e.g., x, y, a, b).

Constant: A fixed numerical value (e.g., 2, -5, 3.14).

Coefficient: The numerical factor of a term containing a variable (e.g., in 3x, 3 is the coefficient).

Like Terms: Terms that have the same variable(s) raised to the same power (e.g., 3x and -7x are like terms, 2y² and 5y² are like terms).

Simplifying Algebraic Expressions: Simplifying involves combining like terms and using the distributive property to remove parentheses.

Combining Like Terms: Add or subtract the coefficients of like terms. For example, 3x + 5x = 8x and 7y - 2y = 5y.

Distributive Property: a(b + c) = ab + ac. This allows us to multiply a term outside the parentheses by each term inside.

Example 1: Simplify the expression: 2(x + 3) - 5x + 1 Distribute: 2(x + 3) = 2x + 6 Rewrite: 2x + 6 - 5x + 1 Combine like terms: (2x - 5x) + (6 + 1) = -3x + 7 Therefore, the simplified expression is -3x +

7. Example 2: Simplify the expression: 3(2y - 4) + y - 2(y + 1)

Distribute: 3(2y - 4) = 6y - 12 and -2(y + 1) = -2y - 2 Rewrite: 6y - 12 + y - 2y - 2 Combine like terms: (6y + y - 2y) + (-12 - 2) = 5y - 14 Therefore, the simplified expression is 5y - 14. 2.2 Factorization: Unraveling the Expression Factorization is the reverse process of expansion (using the distributive property). It involves expressing an algebraic expression as a product of its factors.

Highest Common Factor (HCF): The largest factor that divides into all terms of an expression.

Factorizing by Removing a Common Factor: Identify the HCF of all terms in the expression. Divide each term by the HCF. Write the HCF outside the parentheses and the results of the division inside the parentheses.

Example 3: Factorize: 6x + 9 HCF of 6x and 9 is

3. Divide each term by 3: 6x/3 = 2x and 9/3 = 3 Write the expression as a product: 3(2x + 3)

Example 4: Factorize: 12a² - 8ab HCF of 12a² and -8ab is 4a.

Divide each term by 4a: 12a²/4a = 3a and -8ab/4a = -2b Write the expression as a product: 4a(3a - 2b) 2.3 Difference of Two Squares: An expression in the form a² - b² can be factorized as (a + b)(a - b).

Example 5: Factorize: x² - 16 Recognize the pattern: x² - 16 = x² - 4² (since 16 = 4²)

Apply the formula: (x + 4)(x - 4)

Example 6: Factorize: 9y² - 25 Recognize the pattern: 9y² - 25 = (3y)² - 5² (since 9y² = (3y)² and 25 = 5²)

Apply the formula: (3y + 5)(3y - 5) 2.4 Factorizing Trinomials of the Form x² + bx + c: We need to find two numbers that: Multiply to give c Add to give b Example 7: Factorize: x² + 5x + 6 Find two numbers that multiply to 6 and add to

5. The numbers are 2 and 3 (2 * 3 = 6 and 2 + 3 = 5).

Write the expression in factored form: (x + 2)(x + 3)

Example 8: Factorize: x² - 2x - 8 Find two numbers that multiply to -8 and add to -

2. The numbers are -4 and 2 (-4 * 2 = -8 and -4 + 2 = -2).

Write the expression in factored form: (x - 4)(x + 2) Guided Practice (With Solutions)

Question 1: Simplify the expression: 4(a - 2) + 3a - 5 Solution: Distribute: 4(a - 2) = 4a - 8 Rewrite: 4a - 8 + 3a - 5 Combine like terms: (4a + 3a) + (-8 - 5) = 7a - 13 Therefore, the simplified expression is 7a -

1

3. Commentary: This question tests the distributive property and combining like terms. Make sure to pay attention to the signs when distributing and combining constants.

Question 2: Factorize: 15x²y + 25xy² Solution: Find the HCF: The HCF of 15x²y and 25xy² is 5xy.

Divide each term by the HCF: 15x²y / 5xy = 3x and 25xy² / 5xy = 5y Write as a product: 5xy(3x + 5y) Therefore, the factored expression is 5xy(3x + 5y).

Commentary: This question focuses on identifying and removing the highest common factor. Remember to consider both numerical coefficients and variable powers.

Question 3: Factorize: m² - 49 Solution: Recognize the difference of two squares pattern: m² - 49 = m² - 7² Apply the formula (a² - b² = (a + b)(a - b)): (m + 7)(m - 7) Therefore, the factored expression is (m + 7)(m - 7).

Commentary: Identifying the difference of two squares pattern is key here. Learners need to recognize perfect squares.

Question 4: Factorize: x² + 8x + 15 Solution: Find two numbers that multiply to 15 and add to 8: The numbers are 3 and 5 (3 * 5 = 15 and 3 + 5 = 8).

Write the expression in factored form: (x + 3)(x + 5) Therefore, the factored expression is (x + 3)(x + 5).

Commentary: Focus on systematic finding of factors that satisfy both multiplication and addition constraints. Independent Practice (Questions Only)

Simplify: 3(2x + 1) - 4x + 7 Simplify: 5y - 2(y - 3) + 1 Factorize: 8a - 12 Factorize: 6x² + 9x Factorize: 4p² - 16q² Factorize: y² - 81 Factorize: x² + 6x + 8 Factorize: a² - 5a + 6 Factorize: x² + 3x - 10 Factorize: x² - x - 20