CHAPTER Algebraic Expressins and Fundamental Operatins OBJECTIVES: 1. Algebraic Expressins. Terms. Degree. Gruping 5. Additin 6. Subtractin 7. Multiplicatin 8. Divisin Algebraic Expressin An algebraic expressin is a number, variable r cmbinatin f the tw cnnected b sme mathematical peratin like additin, subtractin, multiplicatin, divisin, expnents, and/r rts. x +, a/5, and 10 - r are all examples f algebraic expressins. Terms An algebraic expressin is ne r mre algebraic terms in a phrase. It can include variables, cnstants, and perating smbls, such as plus and minus signs. It's nl a phrase, nt the whle sentence, s it desn't include an equal sign. x + + 7x + 5 In an algebraic expressin, terms are the elements separated b the plus r minus signs. This example has fur terms, x,,7x, and 5. Terms ma cnsist f variables and cefficients, r cnstants. Variables. In algebraic expressins, letters represent variables. These letters are actuall numbers in disguise. In this expressin, the variables are x and. We call these letters "variables" because the numbers the represent can var that is, we can substitute ne r mre numbers fr the letters in the expressin. Cefficients. Cefficients are the number part f the terms with variables. Inx + + 7x + 5, the cefficient f the first term is. The cefficient f the secnd term is, and the cefficient f the third term is 7. If a term cnsists f nl variables, its cefficient is 1. Cnstants. Cnstants are the terms in the algebraic expressin that cntain nl numbers. That is, the're the terms withut variables. We call them cnstants because their value never changes, since there are n variables in the term that can change its value. In the expressin 7x + x + 8 the cnstant term is "8." Definitin A term cnsists f prducts and qutients f rdinar numbers and letters which represent numbers. Thus 6x, 5x/6, -x 7 are terms. Hwever, 6x + 7x is an algebraic expressin cnsisting f tw terms. A mnmial is an algebraic expressin cnsisting f nl ne term. Thus 7x, x, x / are mnmials. Because f this definitin mnmials are smetimes simpl called terms. A binmial is an algebraic expressin cnsisting f tw terms. Thus x 5x +, x + 6 are binmials. Chapter : Algebraic Expressins and Fundamental Operatins Page 11
A trinmial is an algebraic expressin cnsisting f three terms. Thus x 5x +, x 6 are trinmials. A multinmial is an algebraic expressin cnsisting f mre than ne term. Thus 7x + 6, x + 6x 7x, x + + 10 are multinmial. Plnmial is a mnmial r multinmial in which ever term is integral and ratinal. Fr example x 5x + is a plnmial hwever, are nt plnmial. Degree Gruping The degree f a mnmial is the sum f the expnents f the variables. Such as: 5 x, the degree wuld be 9. Fr x³, the degree wuld be 5. The degree f a plnmial cmes frm the mnmial (term) with the largest degree. 8a b 5 11a b 7b degree 8 5 s the degree f the plnmial is 8. The term with the largest degree is the degree f the whle plnmial. When we have mre cmplex expressins, which ma cmbine several terms, and use multiple peratins, we ma need t grup terms t help sta rganied. Parentheses, ( ), are mst cmmnl used in gruping but u ma als see brackets, [ ], r braces, { }. When a term r expressin is inside ne f these gruping smbls it means that an peratin indicated t be dne n the grup is dne t the entire term r expressin. Fr example, in the sectin n expnents, when a term inside parentheses is raised t a pwer, it means we rise the entire term t that pwer. With terms, this means we raise each numerical cefficient and variable in the term t that pwer. (x) = (x)(x) = x = x Als, when we raise an expressin with parentheses t a pwer, it means t multipl the entire expressin b itself the number f times indicated b the expnent. (5x + ) = (5x + ) (5x + ) (5x + ) (5x + ) Additin When there are multiple gruping smbls in a single expressin, the preferred wa t write them is {[( )]} with the parentheses n the inside, the square-shaped brackets used next, and braces used utermst. An example f an expressin that uses multiple gruping smbls is: {[(x + 6) + x] 1}= 1x + 6 6 Prperties f Additin 1. Cmmutative Prpert f Additin (CPA)-It states that changing the rder f the addends will nt affect the sum.. Assciative Prpert f Additin (APA)-It states that changing the grupings f the addends will nt affect the sum.. Identit Prpert f Additin (IPA)-It states that when u add 0 t an real number, the sum is the number itself. Adding plnmials is just a matter f cmbining like terms, with sme rder f peratins cnsideratins thrwn in. As lng as u're careful with the minus signs, and dn't cnfuse additin and multiplicatin, u shuld d fine. Page 1 Chapter : Algebraic Expressins and Fundamental Operatins
Example 1 Add (x + x x + 5) with (x x + x ) We can add hrintall: (x + x x + 5) + (x x + x ) = x + x x + 5 + x x + x = x + x + x x x + x + 5 = x + 1x x + 1...r verticall: Reserved Subtractin Example Either wa, I get the same answer: x + 1x x + 1. Subtracting plnmials is quite similar t adding plnmials, but u have that pesk minus sign t deal with. Here are sme examples, dne bth hrintall and verticall: Subtract (x 8x 5x + 6) frm (x + x + 5x ) The first thing I have t d is take that negative thrugh the parentheses. Sme students find it helpful t put a "1" in frnt f the parentheses, t help them keep track f the minus sign: Hrintall: (x + x + 5x ) (x 8x 5x + 6) = (x + x + 5x ) 1(x 8x 5x + 6) = (x + x + 5x ) 1(x ) 1 ( 8x ) 1( 5x) 1(6) = x + x + 5x x + 8x + 5x 6 = x x + x + 8x + 5x + 5x 6 = x + 11x + 10x 10 In the hrintal case, u ma have nticed that running the negative thrugh the parentheses changed the sign n each term inside the parentheses. The shrtcut here is t nt bther writing in the subtractin sign r the parentheses; instead, u just change all the signs in the secnd rw. I'll change all the signs in the secnd rw (shwn in red belw), and add dwn: Either wa, I get the answer: x + 11x + 10x 10 Multiplicatin Prperties f Multiplicatin 1. Cmmutative Prpert f Multiplicatin (CPM)-It states that changing the rder f the factrs will nt affect the prduct.. Assciative Prpert f Multiplicatin (APM)-It states that changing the grupings f the factr will nt affect the prduct.. Identit Prpert f Multiplicatin (IPM)-it states that when u multipl an number b 1 the result is the number. Chapter : Algebraic Expressins and Fundamental Operatins Page 1
. Zer Prpert f Multiplicatin (ZPM)- It states that when u multipl an number b 0 the result is 0. There were tw frmats fr adding and subtracting plnmials: "hrintal" and "vertical". Yu can use thse same tw frmats fr multipling plnmials. The ver simplest case fr plnmial multiplicatin is the prduct f tw ne-term plnmials. Fr instance: Example Divisin Example Multipl (5x )( x ) (5x )( x ) = 10x 5 The next step up in cmplexit is a ne-term plnmial times a multi-term plnmial. Fr example: Multipl x(x x + 10) x(x x + 10) = x(x ) x( x) x(10) = 1x + x 0x The next step up is a tw-term plnmial times a tw-term plnmial. This is the simplest f the "multi-term times multi-term" cases. There are actuall three was t d this. Since this is ne f the mst cmmn plnmial multiplicatins that u will be ding, I'll spend a fair amunt f time n this. Multipl (x + )(x + ) (x + )(x + ) = (x + )(x) + (x + )() = x(x) + (x) + x() + () = x + x + x + 6 = x + 5x + 6 If u're dividing a plnmial b smething mre cmplicated than just a simple mnmial, then u'll need t use a different methd fr the simplificatin. That methd is called "lng (plnmial) divisin", and it wrks just like the lng (numerical) divisin u did back in elementar schl, except that nw u're dividing with variables. Divisin f a mnmial b a mnmial: Divide, 18x b x. 18x x 6x Divisin f plnmial b a plnmial: If u're dividing a plnmial b smething mre cmplicated than just a simple mnmial, then u'll need t use a different methd fr the simplificatin. That methd is called "lng (plnmial) divisin", and it wrks just like the lng (numerical) divisin u did back in elementar schl, except that nw u're dividing with variables. Chapter : Algebraic Expressins and Fundamental Operatins Page 1
Example 5 Divide x 9x 10 b x + 1 = = x - 10 First, set up the divisin. Fr the mment, ignre the ther terms and lk just at the leading x f the divisr and the leading x f the dividend. If we divide the leading x inside b the leading x in frnt we will get an x. S put an x n tp. Nw take that x, and multipl it thrugh the divisr, x + 1. First, multipl the x (n tp) b the x (n the "side"), and carr the x underneath and then multipl the x (n tp) b the 1 (n the "side"), and carr the x underneath. Nw subtract x + x frm x 9x 10. We will get 10x 10. Again we have t fllw the same prcess. Divide 10x b x we will get 10. Put this 10 n tp after x. Nw multipl 10 with x + 1 and write the prduct 10x 10 underneath. Finall deduct 10x 10 frm 10x 10. The result will be er. Skill Building In prblem 1 10, write each statement using smbls. 1. The sum f and is the prduct f and. The sum f and equal 5.. The prduct f and is the sum f 1 and.. The difference f x less equals 6. 5. The qutient x divided b is 6. 6. The prduct f 5 and equals 10. 7. The sum f and is the sum f and. 8. The prduct f and x is the prduct f and 6. 9. The difference less equals 6. 10. The qutient divided b x is 6. In prblems 11, evaluate each expressin. 11. 9 1. 5 8 1. 6. 1. 8. 1 1 15. 16. 17. 6 [.5 ( )] 1 8 10 18. ( 5) 8. 1 19. ( 5 ) 0. 1. 5 5 1 1 7... 5 0 15 5 10 In Prblems 5 0, use the Distributive Prpert t remve the parentheses. 1 5. 6( x ) 6. x x 7. x 8. ( x )( x ) 9. ( x )( x 1) 0. ( x 8)( x ) Chapter : Algebraic Expressins and Fundamental Operatins Page 15
In Prblems 1, add the algebraic expressins in each f the fllwing grups. x x 5x ab c 1. x x. c ab x x x x x x c ab In Prblems, subtract the fllwing expressins. a b c d x ab a x.. c a d b x ab a ab In Prblems 5 8 find the indicated prduct f the algebraic expressins. 5. ( x x 9)( x ) 6. ( x x x x )( x ) 7. ( x )(x ) In Prblems 8 0, perfrm the indicated divisins. x x 16 1x 8. 9. x x 0. x x x 1 x Critical Thinking Exercise Writing in Mathematics In Prblems 1 50, state the name f the prpert illustrated. 1. 6 ( ) ( ) 6. 11 (7 ) 11 7 11. 6 ( 7) (6 ) 7. 6 ( ) (6 ) 5. ( ) ( 5) ( 5) ( ) 6. 7 (11 8) (11 8) 7 7. ( 8 6) 16 1 8. 8( 11) ( 88) 1 9. ( x ) 1, x ( x ) 50. ( x ) [ ( x )] 0 51. What is an algebraic expressin? Give an example with ur explanatin. 5. Wh is (x+7) x nt simplified? What must be dne t simplif the expressin? 5. Explain t a friend hw the distributive prpert is used t justif the fact that x + x = 5x. 5. Explain t a friend wh +. = 1, whereas ( + ). = 0. 55. In subtractin cmmutative? Supprt ur cnclusin with an example. 56. In subtractin assciative? Supprt ur cnclusin with an example. 57. Is divisin cmmutative? Supprt ur cnclusin with an example. 58. Is divisin assciative? Supprt ur cnclusin with an example. 59. If = x, wh des x =? 60. If x = 5, wh des x + x = 0? Chapter : Algebraic Expressins and Fundamental Operatins Page 16