Section : Basic Aebra Aebra ike arithmetic deas with numbers Both subjects empoy the fundamenta operations of addition, subtraction, mutipication, division, raisin to a power and takin a root In both, the same symbos are used to indicate these operations (e, + is used for addition, for mutipication etc) and the same rues overn their use In arithmetic we use definite numbers, and we obtain definite numerica resuts when we perform operations on numbers, but in aebra we are mainy concerned with enera epressions and enera resuts in which etters or other symbos are used to represent numbers not named or specified If a etter is used to represent any number from a iven set of numbers, it is caed a variabe A constant is either a fied number such as 5 or, or a etter that represents a fied (possiby unspecified) number Consider the formua for the area A of a circe of radius r: A r Here, A and r are variabes representin the area and the radius of a circe respectivey whie is a constant 1 Aebraic terminooy Any coection of numbers or etters standin for numbers (or powers or roots of these), connected by the sins +, -,, is caed an epression Parts of an epression separated by the sins + or are caed terms An aebraic epression is one of the foowin a constant a variabe a combination formed by performin one or more aebraic operations (addition, subtraction, mutipication, division, raisin to a power or takin a root) on non-zero constants or variabes Eampes of aebraic epressions:, c, c, y y A term is a product or quotient of one or more variabes (or powers or roots of these) and constants Eampes of terms:, y, c, y In a term of the form, is caed the numerica coefficient of and is caed the itera coefficient of When two or more terms (e, -4 and 6 ) have the same itera coefficient, we say that they are ike terms 1
Like terms can aways be combined toether This process of combinin ike terms is caed coectin coefficients 4y + y 10-10 - 4y + y - 7 y An aebraic epression which consists of ony one term is caed a monomia A binomia epression is an aebraic epression consistin of two terms, and a trinomia epression is an aebraic epression consistin of three terms An aebraic epression with two or more terms is caed a mutinomia Epansion and factorization of aebraic epressions In aebra, as in arithmetic, the order in which operations are performed is important Operations are performed from eft to riht Brackets are used to indicate that epressions encosed within them are to be considered as one quantity Thus, operations within brackets are performed first A operations of mutipication and division must be performed before those of addition and subtraction Parenthesis ( ), brackets [ ] and braces { } are used to show the order of performin operation The enera practice is that parentheses are used first and innermost, then brackets, and finay braces Note: 1 When an epression within brackets is preceded by the sin +, the brackets may be removed without makin any chane in the epression Conversey, any part of an epression may be encosed within brackets and the sin + prefied, provided the sin of every term within the brackets remains unatered When an epression within brackets is preceded by the sin -, the brackets may be removed if the sin of every term within the brackets is chaned Conversey, any part of an epression may be encosed within brackets and the sin prefied, provided the sin of every term within the brackets is chaned 1 Mutipyin aebraic epressions In order to mutipy aebraic epressions, we need to appy the aws of indices of section 1 (y )(4 y ) ( 4) ( ) (y y ) 1 1+ y + 1 4 y 5 ( y)(- -5 y 4 z) ( -) ( -5 ) (y y 4 ) z -6 - y 5 z 6 ( y) (6 ) + (6 -y) by the distributive aw 1 5 18 4 y (iv) ( + )( 4) ( + ) ( + )4 by the distributive aw ( ) + ( ) ( 4) ( 4) by the distributive aw + 4 1 1
Factorin aebraic epressions In this section we wi do the reverse of mutipyin aebraic epressions This reverse operation is caed factorin To factor out common terms we appy the distributive property a(b + c) ab + ac in reverse -4y + 1 y 4y(-y) + 4y() 4y(-y + ) y + 6 + 5y + 10 (y + ) + 5(y + ) ( + 5)(y + ) Factorin of trinomias Consider the trinomia Suppose b c where b and c are inteers b c ( + p)( + q) + p + q + pq + (p + q) + pq From the above we see that if we consider a trinomia b c where b and c are inteers, if we can find inteers p and q such that b p + q and c pq, then the trinomia can be factored as b c ( + p)( + q) Now consider the case a Suppose a b c (where a, b and c are inteers with a 0) b c (p + q)(r + s) pr + (ps + qr) + qs In this case the task is to find intera factors p, r of a and q, s of c such that ps + qr b Factor the foowin 7 10 0 4 Soution: 7 10 Here b 7 and c 10 The factors and 5 of 10 are such that their sum + 5 7 Thus 7 10 ( + 5)( + ) (Note: The other pairs of factors of 10 are (10, 1), (-10, -1) and (-, -5), but the sums of these do not add up to 7)
0 Here b -1 and c -0 The factors of -0 (and the sum of the factors) are: 0 and -1 (sum 19) -0 and 1 (sum -19) 4 and -5 (sum -1) -4 and 5 (sum 1) 10 and - (sum 8) -10 and (sum -8) Thus the factors of -0 such that the sum of the factors equas -1 are -5 and 4 Therefore, 0 ( 5)( + 4) 4 ( 4)( + 1) 4 Difference of two squares An epression of the form a is known as the difference of two squares and is factored as foows: a ( a)( + a) 5 Other usefu factorizations y y ( y) y y ( y) y ( y)( y y ) (iv) y ( y)( y y ) Factor the foowin: 9 6y y 16 5 9 0y 5y (iv) 7 Soution: 9 6y y ( + y ) usin factorization above 16 5 (4 5)(4 + 5) difference of two squares 9 0y 5y ( 5y) usin factorization above (iv) 7 ( )( + + 9) usin factorization (iv) above 4
6 Aebraic Fractions The procedures foowed in simpifyin arithmetic fractions can be used to simpify aebraic fractions Simpify the foowin 6 9 6y yz 9 y y 4 16 1 1 1 (iv) Soution: 4 7 4 4 6 9 6y yz 9 ( ) yz 6y ( )( ) ( ) z( ) y y 4 16 ( ) ( 4)( 4) 4 ( 4) y( ) y 1 1 1 1 1 ( 1) 1 (iv) 4 7 4 4 4 7 16 7 7 16 ( 1) 4 4( 1) 4( 1) 4( 1) Formuae One of the most important appications of Aebra is the use of formuae In every form of appied science and mathematics formuae are appied Formuae invove three operations: Construction Manipuation Evauation Eampes of formuae: 5
The formua for the perimeter P of a rectane of enth a and breadth b is P a + b The formua for the area A of a rectane of enth a and breadth b is A ab The formua for the surface area A of a sphere of radius r is A 4r From the above eampes it can be observed that in a formua, one quantity is epressed in terms of other quantities and the formua epresses a reationship between the quantities Consider the formua for the voume of a cyinder with base radius r and atitude h: r h If it is required to epress the atitude h of the cyinder in terms of the voume and base radius r, we woud write h r When one quantity (say h in the above formua) is epressed in terms of other quantities, we ca the reevant quantity the subject of the formua Thus is the subject of the formua h r r h, and h is the subject of the formua The process of transformin one formua into another is caed chanin the subject of the formua Aebraic skis are required to transform one formua into another Evauation is done by substitutin iven vaues for the unknowns The time of vibration t of a simpe penduum is iven by the formua t Make the subject of the formua If v u + fs, make s the subject of the formua and find its vaue when u 15, v 0 and f 5 6
1 Make r the subject of the formua r h and evauate it when 66 and h 7 (Assume that ) 7 Soution: t Therefore t Thus t t Hence 4 v u + fs Therefore fs v u Thus 400 5 175 and f 5, s 17 5 5 10 v u s When u 15, v 0 f 1 r h Therefore r h Thus r h When 66, h 7 we 66 obtain r 9 7 7 4 Evauation of aebraic epressions An epression is evauated by repacin the variabes in the epression with iven numbers and performin the indicated operations By evauatin an epression we obtain a numerica vaue for the epression The vaue of 9 6y y when 1 and y -1 is 9(1) + 6(1)(-1) + (-1) 9 6 + 1 4 7