Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Expand complex algebraic expressions.
• Factorise expressions fully using various techniques.
• Simplify algebraic fractions by factorising.
• Apply factorisation to solve real-world problems.

Lesson summary

Algebraic expressions and factorisation are fundamental building blocks in mathematics. Mastering these concepts is essential not only for success in further mathematics courses but also for developing critical thinking and problem-solving skills that are applicable in various aspects of life. In a South African context, understanding these concepts can empower learners to analyze financial situations, optimize resource allocation, and make informed decisions in various real-world scenarios. For example, understanding how to factorise can help when figuring out how to divide resources like land or money amongst different people fairly.

Lesson notes

Expanding Algebraic Expressions: Expanding algebraic expressions involves removing brackets by applying the distributive property. This property states that a( b + c) = ab + ac. We extend this to more complex expressions by carefully multiplying each term inside the bracket by the term outside.

Example 1: Expanding with a single bracket.

Expand: 3x(2x - 5)

Solution: Multiply 3x by each term inside the bracket: 3x 2x - 3x 5 Simplify: 6x 2 - 15x Example 2: Expanding with multiple brackets.

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

Solution: Use the FOIL (First, Outer, Inner, Last) method or distributive property: x(x - 3) + 2(x - 3)

Expand each term: x 2 - 3x + 2x - 6 Combine like terms: x 2 - x - 6 Example 3: Expanding with negative signs and coefficients.

Expand and simplify: -2(3y - 4) + 5( y + 1)

Solution: Expand each bracket, paying attention to the negative sign: -6y + 8 + 5y + 5 Combine like terms: -y + 13 Factorisation: Factorisation is the reverse process of expanding. It involves expressing an algebraic expression as a product of its factors.

Key factorisation techniques include: Common Factor: Identifying the greatest common factor (GCF) of all terms and factoring it out.

Example 4: Factorise: 4a + 8b Solution: The GCF of 4a and 8b is

4. Factor out 4: 4(a + 2b)

Difference of Two Squares: Recognising expressions in the form a 2 - b 2 , which factorises as (a + b)(a - b).

Example 5: Factorise: x 2 - 9 Solution: Recognise that x 2 is a square and 9 is 3 2 . Apply the difference of two squares formula: (x + 3)(x - 3)

Trinomials: Factoring quadratic trinomials in the form ax 2 + bx + c. For simple trinomials where a = 1, we find two numbers that multiply to c and add up to b.

Example 6: Factorise: x 2 + 5x + 6 Solution: Find two numbers that multiply to 6 and add up to

5. These numbers are 2 and

3. Factorise: (x + 2)(x + 3)

More Complex Trinomials: Factoring quadratic trinomials in the form ax 2 + bx + c where a is NOT

1. We use a process of decomposition: multiply a and c to get a number. Find factors of that number which sum to b. Use these factors to split the bx term. Then factor in pairs.

Example 7: Factorise: 2x 2 + 7x + 3 Solution: Multiply 2 x 3 = 6 Find factors of 6 which add to

7. The factors are 6 and

1. Rewrite the expression: 2x 2 + 6x + x + 3 Factor in pairs: 2x(x + 3) + 1(x + 3)

Factor out the common bracket: (x + 3)(2x + 1)

Simplifying Algebraic Fractions: To simplify algebraic fractions, factorise both the numerator and denominator (if possible) and then cancel any common factors.

Example 8: Simplify: ( x 2 - 4) / ( x + 2)

Solution: Factorise the numerator using the difference of two squares: ( x + 2)(x - 2) / ( x + 2) Cancel the common factor (x + 2): x - 2 Guided Practice (With Solutions)

Question 1: Expand and simplify: 4(2x - 3) - 2( x + 1)

Solution: Expand the first bracket: 8x - 12 Expand the second bracket: -2x - 2 Combine like terms: 8x - 2x - 12 - 2 Simplify: 6x - 14

Commentary: This question tests the ability to expand brackets and combine like terms, including dealing with negative signs.

Question 2: Factorise fully: 6a 2 b - 9ab 2 Solution: Identify the GCF of 6a 2 b and 9ab 2 . The GCF is 3ab.

Factor out the GCF: 3ab(2a - 3b)

Commentary: This question focuses on identifying and extracting the greatest common factor.

Question 3: Factorise: x 2 - 8x + 15 Solution: Find two numbers that multiply to 15 and add up to -

8. These numbers are -3 and -

5. Factorise: (x - 3)(x - 5)

Commentary: This question involves factoring a simple trinomial where a =

1. Question 4: Simplify: ( x 2 - 1) / (x 2 + x)

Solution: Factorise the numerator using the difference of two squares: (x + 1)(x - 1) Factorise the denominator by taking out the common factor x: x(x + 1) Cancel the common factor (x + 1): (x - 1) / x

Commentary: This question combines factorisation techniques with simplification of algebraic fractions.

Question 5: Factorise: 3x 2 + 10x + 8 Solution: Multiply 3 x 8 = 24 Factors of 24 which sum to 10 are 6 and

4. Rewrite the expression: 3x 2 + 6x + 4x + 8 Factor in pairs: 3x(x + 2) + 4(x + 2)

Factor out common bracket: (x + 2)(3x + 4)

Commentary: This question focuses on factorising a more complex trinomial. Independent Practice (Questions Only)

Expand and simplify: -5( a - 2) + 3(2a + 1)

Factorise fully: 12m 3 + 18m 2 - 6m Factorise: y 2 + 2y - 24 Simplify: (2x 2 + 4x) / ( x 2 - 4)

Expand and simplify: ( p + 4)( p - 2) - 3p Factorise: x 2 - 16y 2 Simplify: ( a 2 + 5a + 6) / ( a 2 + 4a + 4)

Factorise fully: 4x 2 - 12x + 9 Factorise: 6x 2 + x - 2 Expand and simplify: ( x + 3) 2 - ( x - 2) 2