Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Expand algebraic expressions involving brackets.
• Factorize by taking out a common factor.
• Factorize the difference of two squares.
• Factorize trinomials.
• Simplify expressions using these techniques.

Lesson summary

Algebraic expressions are the building blocks of algebra, and a solid understanding of them is crucial for success in higher-level mathematics, science, and even everyday financial literacy. They allow us to represent unknown quantities and relationships mathematically, enabling us to solve problems efficiently. This week, we will build upon the concepts introduced last week (terms, coefficients, constants, and like terms) and delve into more complex manipulations of algebraic expressions, focusing on simplification through expansion and factorization.

Lesson notes

2.1 Expansion of Algebraic Expressions Expansion involves removing brackets from an algebraic expression by multiplying each term inside the bracket by the term outside. This is based on the distributive property: a(b + c) = ab + ac.

Example 1: Expanding a Single Bracket Expand: 3x(2x - 5)

Step 1: Multiply 3x by the first term inside the bracket (2x). 3x * 2x = 6x 2 Step 2: Multiply 3x by the second term inside the bracket (-5). 3x * (-5) = -15x Step 3: Combine the results. 6x 2 - 15x Therefore, 3x(2x - 5) = 6x 2 - 15x Example 2: Expanding Two Binomials (Double Brackets)

Expand: (x + 2)(x - 3) We can use the FOIL method (First, Outer, Inner, Last) or the distributive property applied twice. Let’s use the distributive property: Step 1: Distribute the first term of the first bracket (x) over the second bracket (x - 3). x(x - 3) = x 2 - 3x Step 2: Distribute the second term of the first bracket (2) over the second bracket (x - 3). 2(x - 3) = 2x - 6 Step 3: Combine the results. x 2 - 3x + 2x - 6 Step 4: Simplify by combining like terms (-3x + 2x = -x). x 2 - x - 6 Therefore, (x + 2)(x - 3) = x 2 - x - 6 Example 3: Expanding with a Constant Outside the Brackets Expand: 2(3x + 1)(x - 4)

Step 1: Expand the binomials (3x + 1)(x - 4) first.

Using the distributive property: 3x(x - 4) = 3x 2 - 12x 1(x - 4) = x - 4 Combining: 3x 2 - 12x + x - 4 = 3x 2 - 11x - 4 Step 2: Multiply the result by 2. 2(3x 2 - 11x - 4) = 6x 2 - 22x - 8 Therefore, 2(3x + 1)(x - 4) = 6x 2 - 22x - 8 2.2 Factorization of Algebraic Expressions Factorization is the reverse of expansion. It involves writing an algebraic expression as a product of its factors. 2.2.1 Taking Out a Common Factor This involves identifying the highest common factor (HCF) of all the terms in the expression and factoring it out.

Example 1: Factorize: 6x 2 + 9x Step 1: Find the HCF of 6 and 9, which is

3. Step 2: Find the HCF of x 2 and x, which is x.

Step 3: The HCF of the entire expression is 3x.

Step 4: Divide each term in the original expression by 3x and write the result in brackets. (6x 2 / 3x) + (9x / 3x) = 2x + 3 Step 5: Write the original expression as the product of the HCF and the expression in brackets. 3x(2x + 3) Therefore, 6x 2 + 9x = 3x(2x + 3) 2.2.2 Difference of Two Squares An expression in the form a 2 - b 2 can be factorized as (a + b)(a - b).

Example 1: Factorize: x 2 - 16 Step 1: Recognize that x 2 is a perfect square (x x) and 16 is a perfect square (4 * 4).

Step 2: Apply the difference of two squares formula: (x + 4)(x - 4) Therefore, x 2 - 16 = (x + 4)(x - 4) 2.2.3 Factorizing Trinomials A trinomial is an expression with three terms, typically in the form ax 2 + bx + c.

Example 1: Factorize: x 2 + 5x + 6 Step 1: Find two numbers that multiply to give the constant term (6) and add up to give the coefficient of the x term (5). In this case, the numbers are 2 and 3 (2 * 3 = 6 and 2 + 3 = 5).

Step 2: Write the trinomial as the product of two binomials using these numbers. (x + 2)(x + 3) Therefore, x 2 + 5x + 6 = (x + 2)(x + 3)

Example 2: Factorize: 2x 2 + 7x + 3 Step 1: Multiply the coefficient of x 2 (2) by the constant term (3): 23 = 6 Step 2: Find two numbers that multiply to 6 and add up to

7. These numbers are 6 and

1. Step 3: Rewrite the middle term (7x) as the sum of 6x and 1x: 2x 2 + 6x + x + 3 Step 4: Factor by grouping: From 2x 2 + 6x, we can factor out 2x: 2x(x + 3) From x + 3, we can factor out 1: 1(x + 3)

Step 5: Notice that both terms now have a common factor of (x + 3).

Factor that out: (x + 3)(2x + 1) Therefore, 2x 2 + 7x + 3 = (x + 3)(2x + 1) 2.3 Simplifying Algebraic Expressions Simplifying involves combining expansion and factorization techniques to write an expression in its simplest form. This often means removing brackets and combining like terms, or factorizing to cancel common factors. Guided Practice (With Solutions)

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

Solution: Step 1: Expand the first bracket: 2(x - 3) = 2x - 6 Step 2: Expand the second bracket: 3(2x + 1) = 6x + 3 Step 3: Combine the expanded expressions: 2x - 6 + 6x + 3 Step 4: Combine like terms: (2x + 6x) + (-6 + 3) = 8x - 3 Therefore, 2(x - 3) + 3(2x + 1) = 8x -

3. This problem reinforces expanding single brackets and combining like terms, crucial skills for all learners.

Question 2: Factorize completely: 4x 2 - 20x Solution: Step 1: Identify the HCF of 4x 2 and -20x. The HCF of 4 and 20 is 4, and the HCF of x 2 and x is x.

Therefore, the overall HCF is 4x.

Step 2: Divide each term by 4x: (4x 2 / 4x) - (20x / 4x) = x - 5 Step 3: Write the expression as the product of the HCF and the bracketed expression: 4x(x - 5) Therefore, 4x 2 - 20x = 4x(x - 5). This question reinforces the ability to identify and factor out the highest common factor.

Question 3: Factorize: x 2 - 9y 2 Solution: Step 1: Recognize this as a difference of two squares. x 2 is the square of x, and 9y 2 is the square of 3y.