Algebra: Factorising and Long Division 1
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1 Algebra: Factorising and Long Division Mathematics Worksheet This is one of a series of worksheets designed to help you increase your confidence in handling Mathematics. This worksheet contains both theory and eercises which cover:-. Revision of multiplying brackets. Factorisation. Long Division in Algebra There are often different ways of doing things in Mathematics and the methods suggested in the worksheets may not be the ones you were taught. If you are successful and happy with the methods you use it may not be necessary for you to change them. If you have problems or need help in any part of the work then there are a number of ways you can get help. For students at the University of Hull Ask your lecturers. You can contact a Mathematics Tutor from the Skills Team on the shown below. Access more Maths Skills Guides and resources at the website below. Look at one of the many tetbooks in the library. Web: skills@hull.ac.uk
2 Background This Worksheet assumes that you know how to multiply brackets out. There are a number of different ways, some of which are given below. The final method given is used to help with factorisation and long division later in the worksheet. See also Algebra.. Multiplying brackets Multiplying ( y) and ( a b) together can be done in a number of ways (a) The eye-brows method Collecting the terms gives ( + y) (a + b) ( y)( a b) a ya b yb (b) Taking ab,, and yas lengths we have a rectangle ( y) by ( a b) y a a ya From the diagram we see that the total area is given by ( y)( a b) This is equivalent to adding all the bits together giving b b yb ( y)( ab) a ya b yb (c) Use a table y a a ya ( y)( ab) a ya b yb b b yb this method is especially useful when some of the terms are negative or the brackets contain more than terms, see eample (c) below, and also for factorisation and long division. Eamples Epand and simplify the following (a) ( )( ) - - Collecting terms gives ( )( )
3 (b) ( y)( y) y y y y y Collecting terms gives ( y)( y) y (c) y y y y 9 y y 8y y - 0 0y - = 8y y y Hence y y (d) This could be set out as: - - Hence + (e) Hence 0 8
4 Factorisation We can also use the table method to help us factorise quadratic epressions (see also Algebra ): Eamples (a) Factorise This leads to: A B C To get the term A and C must be From the signs both B and D are positive; D hence to get +, B= & D= or vice-versa This gives and in the other two boes a total -term of as required. Solution Factors of are ( )( ) (b) Factorise A B C To get the term take A =, C = D From the signs both B and D are negative, hence to get the +, B = -, D = - or vice-versa This leads to EITHER - This gives and in the other two boes, - a total of which is not correct OR - This gives and in the other two boes, - a total of as required Solution Factors of are ( )( ). (c) Factorise A B C To get take A = & C = D - To get the, B = & D = - or vice-versa or
5 B = - & D = or vice-versa This gives possibilities ( -term total ) ( -term total ) ( -term total ) ( -term total ) From factors of are ( )( ) In practice there is no need to work out and once you ve found that gives the correct result, ecept as a check. (d) Factorise First we notice the common factor of giving We now need to factorise A B C To get the term take A = & C = or A = & C = D - To get the, B = & D =- or B = & D =- or B = & D =- or vice-versa with both number and signs! There are a lot of possibilities here but as the epression has no common factor then the brackets cannot have a common factor, ie we cannot have a factor which has a common factor of (see Algebra for more details.) Trial and error or sheer persistence gives the solution as Eercise Factorise the following d d 0. y. a a...
6 . Division We know that 9, from this we can see that 9 rem or. In eample (d) on page we showed that ( )( ) In the same way, from this, we have: or rem As In the same way rem or We can use the matri method to help with algebraic division. It seems very cumbersome when written out in the amount of detail shown. In practice you should be able to cut down on a lot of the steps. Note, if there is a remainder when dividing by, say, ( ) you can only get a number in the remainder, when dividing by, say, ( ) you can only get, at most, an epression of the form A B in the remainder, In general the maimum power in the remainder will be one less than in the dividend (the term you are dividing by). Eamples (a) Simplify. Let A B remainder S then ( )( A B) S This will come from a matri such as: A B P +S The, - and can be put in giving - Q R A = and, hence, Q = - B P +S The term is so P P =
7 - R hence B = ; R = so we get +S The constant term is - so S - = - S =. (b) Simplify Let A B C remainder then A B C + D D This comes from the following matri A B C P Q +D The, - and - R S T A = and, hence, R = - B C can be put in giving P Q +D term: P P = - - S T hence B = -; S = C Q +D term: Q Q =, - T hence C =, T = D constant term: - + D = D = Solution, Remainder or
8 (c) Simplify Dividing by will give terms starting with leave a possible remainder having an -term and a number. Let B C D remainder E F Then B C D+ E F and - B C D P T Q R E The,, - & can be put in, U V F hence the other terms in column. X Y Z consider term P P=, hence B = ; T = ; X = - C D Q R - Y Z E term gives U V F hence C = 0; U = 0; Y = 0 Q 0 Q = D 0 R 0-0 E term gives E E = 9 0 F constant term F + = F = 0 E term gives R R = - 0 V +F hence D = -; V = -; Z = 0 Z 9 Solution An alternative way of tackling the last eample is:
9 8 let F E D C B Then F E D C B and you can equate the coefficients of to find the values of B, C, D, E and F. When you write the epansion this way it is not so easy to see the coefficients but is good practice, especially for those going on to use Partial Fractions etc. The work is virtually the same as in the first method. Eercise Find the quotients and remainders in the following Answers Eercise y y a a d d
10 Eercise..... Rem Rem - Rem Rem 8 Rem Rem 0 9 Rem Rem Rem We would appreciate your comments on this worksheet, especially if you ve found any errors, so that we can improve it for future use. Please contact the Maths tutor by at skills@hull.ac.uk Updated nd June 0 The information in this leaflet can be made available in an alternative format on request. Telephone
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