Lesson Notes By Weeks and Term v5 - Grade 10
Revision – Week 7 focus
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Revision – Week 7 focus
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Subject: Mathematics
Class: Grade 10
Term: Term 4
Week: 7
Theme: General lesson support
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This week's focus is on revision of key concepts covered thus far in Term
1. Specifically, we will be revisiting algebraic expressions, factorization, and simplification. These skills are absolutely crucial for success in mathematics, not just in Grade 10, but throughout your schooling and in many aspects of daily life. For instance, understanding algebraic expressions helps in budgeting, calculating loan repayments, or even figuring out the best deals when shopping at your local Pick 'n Pay or Shoprite. If you want to become an engineer, accountant, scientist, or even a skilled tradesperson, these foundational algebraic skills will be indispensable.
Let's delve into the core concepts we'll be revising.
A. Factorization: Factorization is the reverse process of expansion. It's about breaking down an algebraic expression into its constituent factors.
Common Factor: Look for the greatest common factor (GCF) among all terms and factor it out.
Example 1: Factorize 6x + 9y. The GCF of 6 and 9 is
3. So, we have: 6x + 9y = 3(2x + 3y)
Example 2: Factorize 12ab - 18ac The GCF of 12 and 18 is 6, and both terms have 'a' in common. 12ab - 18ac = 6a(2b - 3c)
Grouping: When you have four terms, try grouping them in pairs and look for a common factor within each pair.
Example 3: Factorize ax + bx + ay + by.
Group the terms: (ax + bx) + (ay + by)
Factor out the common factors: x(a + b) + y(a + b) Notice that (a + b) is a common factor.
Factor it out: (a + b)(x + y)
Difference of Two Squares: This applies when you have two perfect squares separated by a minus sign. The formula is a² - b² = (a + b)(a - b).
Example 4: Factorize x² - 16 x² is a perfect square, and 16 (4²) is also a perfect square. x² - 16 = (x + 4)(x - 4)
Example 5: Factorize 9p² - 49q² 9p² = (3p)² and 49q² = (7q)² 9p² - 49q² = (3p + 7q)(3p - 7q) Trinomials (x² + bx + c): Find two numbers that add up to 'b' and multiply to 'c'.
Example 6: Factorize x² + 5x + 6 We need two numbers that add up to 5 and multiply to
6. These numbers are 2 and 3. x² + 5x + 6 = (x + 2)(x + 3) Trinomials (ax² + bx + c): This is slightly more complex. You can use trial and error, or the "decomposition" method. Decomposition Method
Example: Factorize 2x² + 7x + 3 Multiply 'a' and 'c': 2 * 3 = 6 Find factors of 6 that add up to 7: 6 and 1 Rewrite the middle term: 2x² + 6x + x + 3 Factor by grouping: 2x(x + 3) + 1(x + 3)
Factor out the common bracket: (x + 3)(2x + 1)
B. Simplifying Algebraic Fractions: Simplifying algebraic fractions involves factoring both the numerator and the denominator (if possible) and then cancelling out common factors.
Example 7: Simplify (x² - 4) / (x + 2) Factor the numerator (difference of two squares): (x + 2)(x - 2) / (x + 2) Cancel the common factor (x + 2): (x - 2)
Example 8: Simplify (x² + 3x + 2) / (x² + 4x + 3)
Factor both the numerator and denominator: Numerator: x² + 3x + 2 = (x + 1)(x + 2)
Denominator: x² + 4x + 3 = (x + 1)(x + 3) So, the fraction becomes: [(x + 1)(x + 2)] / [(x + 1)(x + 3)] Cancel the common factor (x + 1): (x + 2) / (x + 3) Important
Note: You can only cancel factors, not individual terms! For example, you cannot cancel the 'x' in (x + 2) / (x + 3). Guided Practice (With Solutions)
Question 1: Factorize completely: 5x² - 20 Solution: Identify the common factor: 5 Factor out the common factor: 5(x² - 4)
Recognize the difference of two squares: 5(x + 2)(x - 2)
Commentary: Remember to look for a common factor before attempting any other factorization method. This makes the expression simpler to work with.
Question 2: Factorize completely: 3ax + 6ay - bx - 2by Solution: Group the terms: (3ax + 6ay) + (-bx - 2by)
Factor out common factors from each group: 3a(x + 2y) - b(x + 2y)
Factor out the common binomial factor: (x + 2y)(3a - b)
Commentary: Pay close attention to signs when factoring by grouping. The negative sign in front of 'bx' is crucial for getting the correct factors.
Question 3: Simplify: (x² - 9) / (2x + 6)
Solution: Factor the numerator (difference of two squares): (x + 3)(x - 3)
Factor the denominator: 2(x + 3)
Rewrite the fraction: [(x + 3)(x - 3)] / [2(x + 3)] Cancel the common factor (x + 3): (x - 3) / 2
Commentary: Always factorize both the numerator and denominator before attempting to simplify. This allows you to identify common factors easily.
Question 4: Simplify: (x² + 5x + 4) / (x² + x - 12)
Solution: Factor the numerator: x² + 5x + 4 = (x + 1)(x + 4)
Factor the denominator: x² + x - 12 = (x + 4)(x - 3)
Rewrite the fraction: [(x + 1)(x + 4)] / [(x + 4)(x - 3)] Cancel the common factor (x + 4): (x + 1) / (x - 3)
Commentary: Practice your trinomial factorization! It's a skill that will be used repeatedly. Ensure that after factoring each part, you rewrite it carefully and only cancel factors, never terms. Independent Practice (Questions Only)
Factorize: 8x - 12y Factorize: p² - 25 Factorize: 2x² + 8x + 8 Factorize: ab + ac + 2b + 2c Factorize: 3x² - 12 Simplify: (x² - 1) / (x - 1)
Simplify: (x² + 4x + 3) / (x + 3)
Simplify: (2x² - 5x - 3) / (x - 3)
Simplify: (4x² - 9) / (2x² + x - 3)
Simplify: (x³ - x) / (x² + x)