Basic Algebra CAPS Mathematics 1
Outcomes for this TOPIC In this TOPIC you will: Revise factorization. LESSON 1. Revise simplification of algebraic fractions. LESSON. Discuss when trinomials can be factorized. LESSON 3.
Lesson 1 Factorization CAPS Mathematics 3
What is Factorization? Factorization and Product Expansion are reverse processes Product Expansion: a bc d ac ad bc bd Use FOIL Rule Factorize: ac ad bc bd a bc d Product : a bc d e ac ad ae bc bd be Factorize: ac ad ae bc bd be a b c d e 4
Methods of Factorization Common factor Difference between two squares Trinomials (Perfect square) Sum of and difference between two cubes Grouping 5
Common Factor Common factor(s) must occur in each term Must be the HCF Examples ab c 3d 1) ab ac 3ad Highest Common Factor a is the HCF 3 ) 9a x 6a xy 3a a x 3 x y 3 axis the HCF y x 3) x x y 4y y x x x 4y y x y x y Preliminary algebraic manipulation x y is the HCF 6
Difference between two squares Two terms only Terms separated by a minus Terms both perfect squares x - y = x + yx - y Examples 1) 4a 9b ) a 3b a 3b 4 9x y 16 5 4 4 3) x y 3 3 x y x y 4 5 4 5 x y x y Check: Use FOIL-Rule Apply again x y x y x y 4x x y x 4) 4x x y ( x) x 4x y x x y x y Common Factor 7 Difference of Squares
The Factorization of Trinomials What is important Framework Factor Combination Placement of selected factor combination Examples 1) 6a 19ab 10b ) 6a 16ab 10b 3) 6x 59xy 10y 4) 6x 11xy 10y Factor Combinations 6 = 3 = 1 6 and 10 = 10 1= 5 3a b a 5b Common factor! a b 3a 5b x 10y 6x y 3x y x 5y Frameworks 1)a +b a +b )a - b a - b 3) x + y x - y 35 19 53 16 610 11 59 8 35 11
Perfect Square Special type of Trinomial First and last terms are perfect squares Middle term = Twice the product of square roots of first and last terms Two identical factors a ab b a b a b a b Examples 1) 4a 1ab 9b a 3b a 3b = a 3b ) 16x 4 40x y 5y 4x 5y 9
ACTIVITY: Factorization (Part 1) Factorize the following expressions completely: 1) x x 1 6x 1 x ) 16x 81y 4 1 3) 16 p 4q q 4 4) x xy y x y 1 5) a m n a m n n m 6) 3x 3y 6 6 REVISION FOR TEST Attempt examples on your own. Compare your solutions to the solutions given in the next slide. Repeat this procedure a few times before the next test. Consult text-books for additional examples. Remember practice makes perfect! 1
ACTIVITY: Factorization (Part 1) Suggested Solutions 1) x x 1 6x 1 x x x x 1 6x x 1 x x 1 1 3 ) 16x 81y 4 1 4x 9y 6 4x 9y 6 4x 9y 6 x 3y 3 x 3y 3 3) 16 p 4q q 4 16 p q 4q 4 16 p q 4p q 4p q 4) x xy y x y 1 x y x y 1 x y 1 a m n am n m n 5) m na a 1 8 ( Not a perfect square) 6) 3x 3y 6 6 3 x y 6 6 4 4 3 x y x x y y 4 4 3 x y x y x x y y OR 3x 3 y 3 x 3 y 3 13 3 x y x xy y x y x xy y
ACTIVITY: Factorization (Part ) Factorize the following expressions completely: 3 3 7) 8 a b a b 3 8) x x xy xz z y 9) 4 3a b 9 4a 3b 4 10) 5 1 9 6 y m m 11) x1 x 1 MAKE THIS YOUR MOTTO! Attempt examples on your own. Compare your solutions to the solutions given in the next slide. Repeat this procedure a few times before the next test. Consult text-books for additional examples. Remember practice makes perfect! 14
3 3 7) 8 ACTIVITY: Factorization (Part ) Suggested Solutions a b a b a b4a ab b a b a b4a ab b 1 3 8) x x xy xz z y x x y x z x x x y z a b a b 9) 4 3 9 4 3 a b a b a b a b 3 3 4 3 3 3 4 3 6a b 1a 9b6a b 1a 9b 18a 7b 6a 11b 4 10) 5 19 6 y m m 4 5y 9m 6m 1 m 4 5y 3 1 5y 3m 15y 3m 1 x x 11) 1 1 x x 1 1 x x x 1 1 1 x 1x x 1 x 1 x 3 15
Lesson CAPS Mathematics Algebraic Fractions 16
Working with Algebraic Fractions The aim of this section is to discuss operations with algebraic fractions as well as the relevant procedures and restrictions. The following will come under discussion Simplification of fractions Multiplication and division of fractions Addition and subtraction of fractions Simplification of compound fractions 17
Restrictions linked to Algebraic Fractions If the denominator of an algebraic fraction is zero, the expression is meaningless. No factor in the denominator can thus be zero. Hence the variable(s) in the denominator cannot have certain values. Examples 1) ) 1 4 x3 x5 x x x x x x x 3 ( 3) 4 ( )( ) Restrictions : x 3 and x 5 Restrictions : x and x 18
Use of Brackets The following information about the use of brackets is helpful when working with algebraic expressions: 1.) x y y x.) x y x y 3.) x y y x 4.) x y x y 5.) x y y x 3 3 6.) y x x y 4 4 7.) y x x y 5 5 8.) y x x y 19
Simplification of Fractions Factorize both denominator and numerator Divide both by the same factor (s) or cancel common factors Only non-zero factors can be cancelled!! Example 1 a 5b a10b a 5b a 5b a5b a 5b a 5b Factorise both numerator and denominator. Cancel common factor (s). Preferred format. 0
Simplification of Fractions More examples Example x x6 9 3x x3x 3 x 3 Factorise both numerator and denominator. x 3 Cancel common factor and write in preferred form. Note that we assume that x 3 1
Simplification of Fractions More examples Example 3 ax b x ab ax abx ax ab x b axa b ax x b a x b x b x ba 1 ax x b a 1 ax Factorisation Stage 1. Factorisation Stage. Factorisation Final Stage. Simplification. Cancel common factors.
Step 1: Factorize all numerators and denominators. Step : Cancel common factors. Remember: Multiplication and Division of Fractions When dividing with a fraction rather multiply with the reciprocal. 3
Multiplication and Division of Fractions Example 1 Examples ax 3a x x x x x 4 5 3 ax xx 3 x x x 1 x 3 Factorise all numerators and denominators. ax xx 3 x x x 1 x 3 Cancel common factors. ax xx1 Simplified result. 4
Multiplication and Division of Fractions Example More Examples x 1x 3 x 8x 16 x 4x 8x x 8x x 8 x 4 x 8 x x 8 x x 4 8x x 4 x 4 x 8 x 4 x 8 x x 8 x x 4 8x x 4 x 4 x 8 x x 8 8 Simplified result Division to multiplication. Factorise all numerators and denominators. Cancel common factors. 5
Multiplication and Division of Fractions Example 3 More Examples a 5 b5 a a a a 4 6 5 4 4 b a a ab b a a a 1 a 5 1b 4b a 5 a 1 4b a 1 a a 5 a 1 4b a 1 4b a a 1 a 5 1b a 1 1b Simplified result. Factorise all numerators and denominators. Division to multiplication Cancel common factors. 6
Addition and Subtraction of Fractions Factorize denominators where necessary. Determine LCM of the denominators. Write all the fractions as one fraction with LCM as denominator. Simplify fraction as done previously (If required). 7
Addition and Subtraction Examples Example 1 x x1 6 Lowest Common Multiple LCM(,6) = 6 3 x x1 6 3x 6 x1 6 x 7 6 Write as single fraction with LCM as denominator. Simplify fraction 8
Addition and Subtraction More Examples Example 1 x x 1 1 x x1 Write as single fraction with LCM as denominator. x x1 3 Simplify fraction 9
Addition and Subtraction More Examples Example 3 ab a b 1 a b a b b a ab a b 1 a b a b a b a b a ba b ab a a b b a b a b a b Factorize denominators and determine LCM. Write as single fraction with LCM as denominator. ab a ab ab b a b a b a ba b a ba b a b a ba b Simplify fraction 30
Simplification of Compound Fractions Compound fraction is a fraction in which numerator and/or denominator contain fractions. Inside Denominators are those denominators which form part of fractions in the denominator and/or numerator parts of the compound fraction. The idea is to eliminate all these inside denominators i.e. to change the compound fraction into an ordinary algebraic fraction. Method: Multiply numerator and denominator of the compound fraction with the LCM of all inside denominators. 31
Example 1 Simplify Compound Fractions Examples k m m k k m m k k m m k k m m k mk mk k m k mk m What is LCM of all inside denominators? Multiply numerator and denominator with this LCM k m k m k m k m k m Simplify fraction 3
Example Simplify Compound Fractions More Examples 1 1 x1 x 3 1 x x What is LCM of all inside denominators? Multiply numerator and denominator with this LCM 1 1 x1 x 3 1 x x x x1 x x1 xx1 x x xx Simplify fraction x1 x1 x 3x 3 x x 1 x 1 3 1 1 33
Example 3 1 x 1 x 1 1 x 1 x 1 Simplify Compound Fractions 1 x 1 x 1 1 x 1 x 1 x x 1 1 x x x x 1 1 1 x x 1 1 1 More Examples What is LCM of all inside denominators? Multiply numerator and denominator with this LCM Simplify fraction x x x 11 x x x x x 11 x 3x x 3 34
Tutorial: Algebraic Fractions (Part 1) Simplify the following fractions: 1) ) 3) x y x xy y a ab 4b a 4ab 4b a ab b a b b a x x 1 x 1 x x 1 1 x 1 x x 3 35
Tutorial: Algebraic Fractions (Part 1) Suggested Solutions Simplify: 1) x x y x y x xy x y y y x x y y 36
Tutorial: Algebraic Fractions (Part 1) Suggested Solution Simplify: ) a ab 4b a 4ab 4b a ab b a b b a a ba b a b a b a b a b a b a b a b a b 0 37
3) Tutorial: Algebraic Fractions (Part 1) Suggested Solution Simplify: x x 1 x 1 x 3 x 1 1x 1 x x x x x 1 x x 1 x 1 x1 x x x 1 1 1 1 1 1 x 1 x x 1 x x x x x x x 3 3 x x x x x x x 1 x x 1 1 1 3 x 1 38
Tutorial: Algebraic Fractions (Part ) Simplify the following fractions: 4) 5) 6) 5 3 3 x x x x x 1 1 a 1 4 a 1 1 x x 1 x 1 x 3 3 39
Tutorial: Algebraic Fractions (Part ) Suggested Solutions Simplify: 5 3 4) 3 x x x x x 1 3 x x x x x x 1 1 1 1 x x x 1 3x x x 1 x 1 1 x x x 3x 5x 1 1 x x 1 x 1 x x x 40
41 Tutorial: Algebraic Fractions (Part ) Suggested Solutions Simplify: 1 5) 1 4 1 1 4 1 4 1 1 1 1 a a a a a a a a a a a a a a a
Tutorial: Algebraic Fractions (Part ) Suggested Solution Simplify: 1 1 x 6) x 1 x 1 x 3 3 1 x 1 1 x 1 x x 3x 3 3 3 3 1 1 1 1 x x 3x x x x 1 x x 1 x x 1 x 1 3 x 1 3 x 1 4
End of Basic Algebra REMEMBER! Consult text-books for additional examples. Attempt as many as possible other similar examples on your own. Compare your methods with those that were discussed in the Class. Repeat this procedure until you are confident. Do not forget: END Practice makes perfect! 49