TECHNIQUES IN FACTORISATION

Transcription

1 TECHNIQUES IN FACTORISATION The process where brackets are inserted into an equation is referred to as factorisation. Factorisation is the opposite process to epansion. METHOD: Epansion ( + )( 5) 15 Factorisation Bring all terms to one side of the equation and simplify by applying one or more of the following techniques: (a) Remove the Highest Common Factor. (b) If the polynomial epression consists of TWO terms (binomial epression), factorise by using one of the following rules: The difference of two squares. ( a b)( a b) a b = + The sum or difference of two cubes. ( a + b)( a ab b ) ( a b)( a + ab b ) a + b = + a b = + (c) If the polynomial epression consists of THREE terms (trinomial epression), factorise by using one of the following rules: Rules for perfect squares. ( a ) ( a ) a + a ab + b = b + ab + b = b The FOIL method (to write epressions as linear factors). Completing the Square. Write the equation as a quadratic epression by using substitution (Let A = method). The factor theorem and long division. Note: A quadratic trinomial is an epression of the form: a + b + c. (d) If the polynomial epression consists of FOUR items, factorise using: The factor theorem and long division. Grouping two and two. Grouping three and one. The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 10

2 NUMBER SYSTEMS Algebraic epressions are usually factorised over the following number systems: The rational number system (Q ) The real number system ( R ) The integer number system ( Z ) THE RATIONAL NUMBER SYSTEM (Q) When factorising over Q, the factors can only consist of rational numbers i.e. whole numbers or numbers that may be written as fractions. For eample:, +, + 5, are rational numbers. Whole numbers may also be 5 = 1 classified as rational numbers, as they too can be epressed as a fraction. Eg. Note: Irrational numbers are epressions that cannot be epressed as a fraction. For eample: e, π and numbers that display an infinite number of decimal places. THE REAL NUMBER SYSTEM (R) When factorising over R, the factors can consist of both rational and irrational numbers (surds). For eample: 1,, 10, are real numbers. 5 is not part of the real number system as the square root of a negative number cannot be evaluated. THE INTEGER NUMBER SYSTEM (Z) The integer number system includes positive and negative whole numbers, as well as zero. i.e....,, 1, 0,1,,... The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 11

3 HIGHEST COMMON FACTORS The highest common factor (HCF) is the highest single term that can be taken outside a bracket. The HCF includes the largest common numeric factor (number), and the highest common power of each pronumeral (letter). Common factors should be removed before applying any other technique in factorisation. Not only does the removal of a common factor make the factorisation process simpler, it may also help to reveal what further techniques in factorisation need to be applied. METHOD: Step 1: Identify the numeric factors and letters that are common to all terms. Step : Write the common factor outside a set of brackets. Step : Write the given equation inside the brackets and divide each term by the highest common factor. Step 4: Simplify. QUESTION 6 Factorise 4a + 6ab. Solution Identify the HCF: The highest term that divides into both Write the HCF outside a set of brackets: 4a and 6 ab is a. a ( ) Divide each term by the HCF and simplify: 4a 6ab 4a + 6ab = a + = a(a + b) a a The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 1

4 QUESTION 7 Factorise 10 y 5y + 5 y. Solution The HCF is 5 y. 10y 5y y 5y y = 5y 5y 5y 5 y + 5y = 5y ( y + ) QUESTION 8 1 Factorise +. Solution The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 1

5 THE DIFFERENCE OF TWO SQUARES (DOTS) ( )( ) ( )( ) a b = a + b a b = a+ b a b The Difference of Two Squares (DOTS) is used to factorise equations that consist of the difference of two terms (binomial epressions), both of which are perfect squares. To factorise these epressions: Step 1: Remove the highest common factor. Step : Take the square root of each entire term. Step : Add and subtract each term. Note: The sum of two squares ( a + b ) cannot be factorised. QUESTION 9 Factorise each of the following epressions: (a) a b + + a b = a ± b = a ± b = ( a + b)( a b) (b) 4 16y 4 16y = 4( 4y ) = 4( ± 4y ) = 4( ± y) = 4( + y)( y) (c) 16 7y The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 14

6 (d) ( 1) 4 (e) ( 1) ( + 1) (f) 4( y) 9( + y ) The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 15

7 THE SUM AND DIFFERENCE OF TWO CUBES The sum or difference of cubic binomials (two terms that are cubed) may be factorised using the following rules: Note: The identity for the sum of two cubes is: a + b = ( a + b)( a ab + b ) The identity for the difference of two cubes is: a b = ( a b)( a + ab + b ) a represents the cube root of the first term. b represents the cube root of the second term. The sign in between the two terms determines which formula is to be applied. a ± ab + b cannot be factorised further. The quadratic term ( ) To factorise these epressions: Step 1: Remove the highest common factor. Step : State the epressions for a and b (take the cube root of each given term). Step : Substitute the epressions for a and b into the appropriate rule. QUESTION 10 Factorise the following epressions: (a) This epression can be written in the form of the sum of two cubes: As a + b = ( a + b)( a ab + b ) = () + () = ( + )[() ()() + () = ( + )( ) ] (b) a 15b This epression can be written in the form of the difference of two cubes: As a b = ( a b)( a + ab + b ) a 15b = a (5b ) = + + ( a 5 b)[( a) ( a)(5 b) (5 b) ] = + + ( a 5 b)( a 5ab 5 b ) The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 16

8 (c) ( y) + 50 (d) ( + y) ( y ) The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 17

9 QUADRATIC TRINOMIALS A quadratic trinomial is an epression of the form: a + b + c. There are a variety of methods that may be used to factorise trinomials, including: Rules for perfect squares. Producing two linear factors by trial and error. Completing the Square. PERFECT SQUARES If the given equation is a quadratic trinomial (a quadratic equation consisting of three terms), where one term is two times the product of the square root of the other two terms, then the equation may be factorised using the following rules: ( a ) ( a ) a + a ab + b = b + ab + b = b Note that the sign in front of the term containing the product ( ab ) determines which formula is to be applied. METHOD: Step 1: Arrange terms in order of the power of the variable (usually ). Step : Take the square root of the first term. Step : Carry down the sign in front of the coefficient of the middle term. Step 4: Take the square root of the third term (the constant). Step 5: Arrange the answer in the following format: (square root of 1 st term sign in front of the nd term square root of rd term) ( + ) = ( + ) The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 18

10 QUESTION 11 Factorise the following epressions: (a) (b) 5 + 0y + 4y (c) 9 0y + 5y The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 19

11 FACTORISING QUADRATIC TRINOMIALS USING THE FOIL TECHNIQUE Many quadratic trinomials (quadratic equations consisting of three terms) can be factorised to produce two linear factors. This technique involves the construction of two pairs of brackets, and inserting the appropriate factors by using FOIL in reverse. METHOD: Step 1: Bring all terms to one side of the equation and collect like terms. Step : Arrange terms in order of the power on. Step : Write down the factors of the term involving Step 4: Write down the factors of the term that is independent of (the constant). Step 5: Choose factors in such a way that the sum of the products of the (first term last term) and the (outside term inside term) is equal to the term involving.. For Eample: Factorise + + y y. + + y y = ( )( ) The product of the first two terms represents the term involving. + + y y = ( + y)( + y) The product of the second two terms represents the term that is independent of. Factors must be chosen in such a way that the sum of the products of the (first term last term) and the (outside term inside term) is equal to the term involving. ( First Outside)( Inside Last) This may involve some trial and error! Use the signs in front of the terms in the quadratic equations to help in this process. Note: There are four types of quadratic trinomials each characterised by different patterns in signs. The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 0

12 TYPE 1: TWO PLUS SIGNS For Eample: This plus sign tells you that the signs are positive. 1. This plus sign tells you that the signs are the same in the two brackets and that you ADD the factors to find the middle term. ( 1 = and + 1 = 4) Therefore: = ( + )( + 1) TYPE : ONE MINUS AND ONE PLUS SIGN For eample: This sign tells you that the signs are negative. 1. This plus sign tells you that the signs are the same in the two brackets and that you ADD the factors to find the middle term. ( 5 1 = 5 and = 6) Therefore: = ( 5)( 1) The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 1

13 TYPE : TWO MINUS SIGNS For Eample: 6 7. This minus sign tells you that the biggest factor is negative. 1. This negative sign tells you that the signs in the brackets are different and that you subtract the factors to find the middle term. ( 7 1 = 7 and 7 1 = 6) 6 7 Therefore: 6 7 = ( 7)( + 1) TYPE 4: ONE PLUS AND ONE MINUS SIGN For Eample: + 6. This plus sign tells you that the biggest factor is positive. 1. This negative sign tells you that the signs in the brackets are different and that you subtract the factors to find the middle term. ( = 6 and = 1) + 6 Therefore: + 6 = ( + )( ) The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page

14 QUESTION 1 Factorise the following epressions. (a) p 14 p 15 p 14 p 15 = ( p 15)( p + 1) (b) 10 (c) = ( 7)( 7) = ( 7) (d) t 15t (e) The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page

15 (f) 6(b 1) 14(b 1) + 4 The School For Ecellence 016 Summer School Unit 1 Maths Methods Book 1 Page 4