Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Simplify algebraic expressions.
• Factorise by taking out a common factor.
• Factorise the difference of two squares.
• Factorise trinomials of the form x² + bx + c.
• Apply factorisation to solve simple problems.

Lesson summary

This week, we delve into the fascinating world of algebraic expressions and factorisation. Algebraic expressions are like mathematical sentences that use variables (letters representing unknown numbers), constants, and operations (+, -, ×, ÷). Factorisation is the process of breaking down an algebraic expression into simpler expressions that, when multiplied together, give you the original expression. Think of it like reverse multiplication! This skill is crucial not only for further mathematics but also for problem-solving in various fields, from engineering to finance. Understanding these concepts opens doors to more complex mathematical ideas and strengthens your logical thinking.

Lesson notes

2.1 Understanding Algebraic Expressions: An algebraic expression is a combination of variables, constants, and mathematical operations.

For example: `3x + 5` (x is a variable, 3 and 5 are constants) `2y² - 7y + 1` (y is a variable, 2, 7, and 1 are constants) `ab + c` (a, b, and c are variables)

Terms: Parts of an algebraic expression separated by + or - signs are called terms. In `3x + 5`, `3x` and `5` are the terms.

Like Terms: Terms with the same variable raised to the same power are called like terms. `3x` and `5x` are like terms because they both contain 'x' raised to the power of 1. `2y²` and `7y²` are like terms because they both contain 'y²'.

However, `3x` and `3x²` are not like terms because the powers of 'x' are different.

Combining Like Terms: We can simplify algebraic expressions by combining like terms. This involves adding or subtracting the coefficients (the numbers in front of the variables) of the like terms.

Example: Simplify `3x + 5x - 2x` All terms are like terms (they all have 'x').

Add the coefficients: `3 + 5 - 2 = 6` Therefore, `3x + 5x - 2x = 6x` The Distributive Property: This property states that `a(b + c) = ab + ac`. We use this to expand expressions that involve brackets.

Example: Expand `2(x + 3)` Multiply 2 by each term inside the brackets: `2 x + 2 * 3` Simplify: `2x + 6` 2.2 Factorisation: Factorisation is the reverse process of expansion. It involves finding the factors that, when multiplied together, give the original expression. 2.2.1 Factorisation by Taking Out a Common Factor: This is the most basic type of factorisation. Look for a common factor in all the terms of the expression.

Example: Factorise `6x + 12` The highest common factor (HCF) of 6x and 12 is

6. Divide each term by 6: `6x ÷ 6 = x` and `12 ÷ 6 = 2` Write the expression as the product of the common factor and the remaining terms: `6(x + 2)`

Example: Factorise `4a²b - 8ab² + 12ab` The HCF of `4a²b`, `-8ab²`, and `12ab` is `4ab`. Divide each term by `4ab`: `4a²b ÷ 4ab = a`, `-8ab² ÷ 4ab = -2b`, and `12ab ÷ 4ab = 3` Write the expression as the product of the common factor and the remaining terms: `4ab(a - 2b + 3)` 2.2.2 Factorising the Difference of Two Squares (DOTS): This applies to expressions of the form `a² - b²`. The factorisation is `(a + b)(a - b)`. Remember, it must be a difference (subtraction) of two perfect squares.

Example: Factorise `x² - 9` Recognise that `x²` is the square of `x` and `9` is the square of `3`.

Apply the formula: `(x + 3)(x - 3)`

Example: Factorise `25m² - 16n²` Recognise that `25m²` is the square of `5m` and `16n²` is the square of `4n`.

Apply the formula: `(5m + 4n)(5m - 4n)` 2.2.3 Factorising Trinomials of the Form x² + bx + c: A trinomial is an expression with three terms. For this type of trinomial, we need to find two numbers that: Multiply to give `c` (the constant term) Add to give `b` (the coefficient of the x term)

Example: Factorise `x² + 5x + 6` We need two numbers that multiply to 6 and add to

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

Therefore, the factorisation is `(x + 2)(x + 3)`

Example: Factorise `x² - 8x + 15` We need two numbers that multiply to 15 and add to -

8. The numbers are -3 and -5 (because -3 -5 = 15 and -3 + -5 = -8).

Therefore, the factorisation is `(x - 3)(x - 5)`

Example: Factorise `x² + 2x - 8` We need two numbers that multiply to -8 and add to

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

Therefore, the factorisation is `(x + 4)(x - 2)` Guided Practice (With Solutions)

Question 1: Simplify the expression: `7y - 3y + 2 + 5 - y` Solution: Identify like terms: `7y`, `-3y`, and `-y` are like terms. `2` and `5` are like terms.

Combine like terms: `7y - 3y - y = (7 - 3 - 1)y = 3y` and `2 + 5 = 7` Write the simplified expression: `3y + 7`

Commentary: This question reinforces the concept of identifying and combining like terms, a fundamental skill for simplifying algebraic expressions.

Question 2: Factorise: `10x² - 15x` Solution: Find the HCF: The HCF of `10x²` and `-15x` is `5x`.

Divide each term by the HCF: `10x² ÷ 5x = 2x` and `-15x ÷ 5x = -3` Write the expression as the product of the HCF and the remaining terms: `5x(2x - 3)`

Commentary: This question focuses on factorisation by taking out a common factor. Identifying the correct HCF is crucial.

Question 3: Factorise: `m² - 49` Solution: Recognise the difference of two squares: `m²` is the square of `m` and `49` is the square of `7`.

Apply the formula: `(m + 7)(m - 7)`

Commentary: This question tests the ability to recognize and apply the difference of two squares factorisation pattern.

Question 4: Factorise: `x² + 7x + 12` Solution: Find two numbers that multiply to 12 and add to 7: The numbers are 3 and

4. Write the factorised expression: `(x + 3)(x + 4)`

Commentary: This question requires students to find the correct pair of numbers that satisfy the multiplication and addition conditions for trinomial factorisation.