THE DISTRIBUTIVE LAW ( ) When an equation of the form a b c is epanded, every term inside the bracket is multiplied by the number or pronumeral (letter), and the sign that is located outside the brackets. This rule is known as the Distributive Law. ( b c) = ( a b) ( a c) a = ab ac Note: To avoid mistakes, include arrows above or below the terms that are being multiplied. To epand a quadratic epression we often use FOIL i.e. First, Outside, Inside, Last. ( a b)( c d ) = ac ad bc bd When epanding epressions, we remove brackets, and then simplify by collecting like terms. There is usually some simplifying to do afterwards which includes collecting like terms. This process can be etended to epand three binomial factors i.e. cubic epansions. For eample: ( )( )( ) Epand 1 4. Do not attempt to epand all three brackets at one time. ( 1 )( )( 4 ) = ( )( 4 ) * Epand the first two brackets = = ( 4 )( 4 ) ( 4 16 4 1 ) = 1 1 The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page
QUESTION 1 Epand the following epressions: (a) 5( 7) (b) ( 1)( 5) = 5 1 1 5 = 6 15 5 = 6 1 5 (c) zz ( 1) ( z 5) (d) 5(7 )(5) The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page 4
The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page 5 (e) ) 7 5 ( y (f) ( 7)( )( 5) (g) ( ) ( ) 5 4 ( ) ( ) ( ) ( ) ( ) ( ) 5 4 = 10 6 4 = 14 =
EXPANDING EXPRESSIONS BY RULE Some quadratic and cubic epressions will always follow a certain pattern, and therefore, we can epand these epressions by using a standard set of rules. PERFECT SQUARES Perfect squares are epressions that are written to the power of two. For eample:, ( ) and ( 1). Perfect squares may be epanded directly or by applying the following rules: ( a b) = a ab b ( a b) = a ab b Take care to avoid confusing the following: ( a b) a b ( a b) a b common eam error QUESTION 14 Epand and simplify each of the following epressions: (a) ( a b) = (a) (a)( b) ( b ) = 9a 6ab b (b) ( 5 7y) (c) ( 1 ) The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page 6
THE DIFFERENCE OF TWO SQUARES Given the product of two epressions that consist of the sum and difference of the same terms, we epand by applying the Difference of Two Squares in reverse. ( a b)( a b) = a b QUESTION 15 Epand and simplify the following epressions: (a) ( 7 )(7 ) = (7) () = 49 4 (b) ( 5)( 5) The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page 7
EXPANDING PERFECT CUBES BY RULE Perfect cubes are epressions that are written to the power of three. For eample:, ( py ), ( ), ( 6) a. Perfect cubes may be epanded directly or by applying the following rules: ( a b) = a a b ab b ( a b) = a a b ab b Take care to avoid confusing the following: ( a b) ( a b) a b a b QUESTION 16 Epand the following epressions: (a) ( ) = () ( ) () ()() () = 8 (4 )() ()(9) 7 = 8 6 54 7 (b) ( 8y) The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page 8
QUESTION 17 Epand the following epressions: (a) 4 (b) 15 QUESTION 18 Epand and simplify the following epressions: (a) w (11 4w) (b) (17 y)( 5 y) The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page 9
(c) ( 4w z) (d) ( )( ) (e) 1 1 The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page 0
TECHNIQUES IN FACTORISATION The process where brackets are inserted into an equation is referred to as factorisation. Factorisation is the opposite process to epansion. Epansion ( )( 5) 15 Factorisation METHOD Bring all terms to one side of the equation and simplify by applying one or more of the following techniques: (a) (b) Remove the Highest Common Factor. 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 b)( a ab 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 The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page 1
THE DIFFERENCE OF TWO SQUARES ( )( ) ( )( ) 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 19 Factorise the following epressions: (a) 49 (b) 9 64z ( ) ( 8z) ( 8z) ( 8z) = = (c) 15( ) 15 (d) 16 4 y 6 The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page
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 (FOIL). Completing the Square. FACTORISING QUADRATIC TRINOMIALS BY RULE 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. QUESTION 0 Factorise 6 9. Solution 6 9 9 ( ) = ( ) The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page
QUESTION 1 Factorise the following epressions: (a) 14 49 (b) 4 0 5 (c) 5z 60zy 6y The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page 4
FACTORISING QUADRATIC TRINOMIALS 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. You should already be familiar with this process here are some questions to try, to see if you are already able to do there! QUESTION Factorise 10 1. Solution Write down the factors of the term involving Factors: and : 10 1 = ( )( ) Write down the factors of the term that is independent of (the constant): Factors: 1 1, 1 1, 7, 7 Choose the 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 : ( First Outside)( Inside Last) As the middle term and the last term are positive, both factors of the term that is independent of are positive. 10 1 = ( 7)( ) The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page 5
QUESTION Factorise the following epressions: (a) 7 18 (b) 1 4 (c) 9 18 (d) 9y 48yz 64z The School For Ecellence 016 Summer School Year 11 Mathematics Book 1 Page 6