Math 017 Materials With Exercises

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$Math 171 Review for the Final Exam$

$Math 171 Test 2$

$MATH 171 SYLLABUS$

a x and b 3 , so now we can rewrite the right-hand side of the formula, substituting x in place of2 2a and 3 in place of b . We will write the formula directly above the expression x 3 to make iteasier to see how x must replace a and 3 must replace b on the right hand side.2 2a b ( a b)(a b)x2 3 2 ( x 3)( x 3)Notice, that by applying the Difference of Squares formula we were able to rewrite the expression2 2x 9 in a form of a product, namely x 9 ( x 3)( x 3). Recall that the process of rewriting anexpression as a product is called factorization (we used factorization (*) to simplify algebraicfractions). So far the only method of factorization we learned was a method of factoring a commonfactor (see, Lesson 5). The example above provides us with another method of factorization, a method2 2that can be applied to any expression that can be rewritten to match the form a b . Let ussummarize the above considerations.If we need to factor the expression x2 9 , we must notice that it can be rewritten as a difference of2 2two squares, x 3 and then the Difference of Squares formula can be applied to get the desiredfactorization2x 9 ( x 3)( x 3) ..Example 11.9 Using the formula2 2a b ( a b)(a b), factor the following expressions.2a) 25 xb) x2 192 4c) 36m 49n2d) m2 ( 1 3m)Solution:22 22a) The expression 25 x must be written in the form a b . Since 25 5 ,2 2we have 5 x . We recognize that a 5 and b x , and substitute those values in theright-hand side of the equation( a b)(a b) 5 x 5 x .The expression 2 is factored: 25 x 5 x5 x225 xb) Write the expression2x 2 1 in the form922a b2.. To this end, notice that1b , and31 1 1 2 1 2 1 2 , and thus x x . We recognize that a x ,9 3 39 3substitute those values in the right-hand side of the equation 1 1 ( a b)(a b) x x . 332 1 1 1 Thus x x x .9 33 __________________________(*) We should make sure that we understand that factorization of an expression does not change its “meaning”. Theobtained expression, although written in a different form, is still equal to the original one. In the example, we can convinceourselves that the two expressions are equal by removing parentheses in (x − 3) (x + 3) and collecting like terms.(x − 3) (x + 3) = xx +3x−3x−3∙3 = x² − 9.110
c) Since36 m22m (6 ) ,4 249n (7n)2, thus36mn2 42 2 2 49n (6m) (7 ) .2 2The left-hand side of the equation is written in the form a b with a 6mand2b 7n . Substituting those values in ( a b)(a b)gives us the desired factorization2 42236m 49n (6m 7n)(6m 7n) .2 2d) The expression is already written in the form a b , with a m andb 1 3m . We replace a and b in ( a b)(a b)with m and 1 3m, respectively, andsimplify.( a b)(a b) [ m (1 3m)][m (1 3m)] ( m 13m)(m 13m) (4m1)(1 2m)22. This gives us the desired factorization: m (1 3m) (4m1)(1 2m).<strong>Exercises</strong> with Answers (For answers see Appendix A)Ex.1The expression2 3 y is written in the form 42( y a) . What is the value of a?Ex.2 The following expressions are written in the form ax by . What are the values of a and b inthese representations?2a) 3x 4yb) 4x y3Ex.3 The expression 3 x is writtena) in the form p x . Determine the value of p in this representation.b) in the form p x . Determine the value of p in this representationEx.4The expression25x is written in the forma bx . What are the values of a and b?2Ex.5 The expression ( 7) x is written in the form a 2 x . What is the value of a?Ex.6 The expression 4x 2yis written in the form A 2B. What algebraic expression represents Aand B.2Ex.7 The following expressions are written in the form XY For each of them determine whatalgebraic expression represents X and Y .22a) 3ab b) 3(ab )cEx.8 The following expressions are written in the form . For each of expressions identifyb(without rewriting the expression) what algebraic expression represents c and b .99( x 1)xa)b)92y9111
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Math 017MaterialsWithExercises
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TABLE OF CONTENTSLesson 1Variables
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Lesson 1___________________________
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e) „minus x ‟ or „the opposit
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For example,Evaluate 5 x when x 10
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3xd) x( y)e) 2f) 2x yz w(t)Ex.4 T
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Ex.16 Let x 3. Rewrite the express
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e) x5f) x2Ex.28 Substitute“undefi
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Lesson 2___________________________
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Solution:Please, notice the use of
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g)i)4( a 2)h)8y (x)4(ab)j) ( a d)
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1 2Ex.19 If possible, evaluate when
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Lesson 3___________________________
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Commutative Propertyof Additionx y
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2 x 2 xFor example, and thus als
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Example 3.10 Determine which of the
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Exercises with Answers (For answers
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Ex.16 Write the following expressio
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s) ( 0.2xy)( 0.3z)t)4x1Ex.24 Is 1
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Lesson 4___________________________
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Numerical coefficientsNumerical Coe
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For your convenience, we will displ
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9 818081a) 9 98097 6 76134 4 4 4
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Ex.8Simplify by raising to the indi
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73a 2xxi) 2aj)2( 42a)x(3xk)4x2 ) 3
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Lesson 5___________________________
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FactorizationThe Distributive Law l
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Simplification of algebraic fractio
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Mistake 5.1When factoring 3 from 3x
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Ex.13 Factor2a) 5 a from the follow
Page 62 and 63: 24x ( 5x)g)xa bi)7(a b)bc ( b e)
Page 64 and 65: Adding and subtracting like termsIf
Page 66 and 67: ) This should be written as 4x 2
Page 68 and 69: Ex.4 Circle all terms that are like
Page 70 and 71: 682 122k) 6 ( d a) a dl) 3a1(4a
Page 72 and 73: Lesson 7___________________________
Page 74 and 75: For instance, the area of a rectang
Page 76 and 77: a) a b 2bc) a bd) ae)b)aba 22a a
Page 78 and 79: Ex.24 Express the expressiona) m s
Page 80 and 81: A solution, the solution set, and w
Page 82 and 83: ■ Solve the following equation 3x
Page 84 and 85: sides (subtracting 4 x from both si
Page 86 and 87: Mistake 8.2While solving an equatio
Page 88 and 89: Ex.12 Determine whether the followi
Page 90: Lesson 9___________________________
Page 94 and 95: xy x 1 yFactor x .x( y 1)1yDivide
Page 96 and 97: a) b b) a c) tEx.15. SolveEx.16. So
Page 98 and 99: For each number you should check if
Page 100 and 101: ■ Solve the inequality 4x 9 1.
Page 102 and 103: ■ Solve the following inequality
Page 104 and 105: Ex.10 Graph the following number se
Page 106 and 107: d) z 3e) z 13 Ex.22 Knowing x 1
Page 108 and 109: Example 11.2 The expression( 22 2x
Page 110 and 111: Solution:a) To match the form y mx
Page 114 and 115: Ex.9 For each of the following expr
Page 116 and 117: 114xc) y 5d) y 3x 2 0e) 3y 6x1f
Page 118 and 119: APPENDIX A:Lesson 1ANSWERS TO EXERC
Page 120 and 121: 49C 3255 2(L+W)62mc75a) a b expon
Page 122 and 123: 3(m n)a(x1)g)h)2tx19s s s sThe opp
Page 124 and 125: 52233x1 2 x5 3 1513 a) m ( 2m) 2mb
Page 126 and 127: c) Numerator: one term, 3ab (with t
Page 128 and 129: 18 a) 9 b) 3 c) 3 d) 3e) 6 f) 279
Page 130 and 131: 11 a) x 1 b)12 a) x abcx a b42 b
Page 132 and 133: c) x 2d) a 7e) x 10f) a 225 a)
Page 134 and 135: x or 4x ( 7) 0; a 4; b 7g) x (
Page 136 and 137: 3. Let m = ─3. Evaluate the follo
Page 138 and 139: 1. Write the following statements a
Page 140 and 141: 6. Solve the following equations:8.
Page 142 and 143: 8. Factor the following expressions
Page 144 and 145: ___________________________________
Page 146: 3 ;b) x 3 y 6 xy 2 A xy 28. a) x
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Math 017 Materials With Exercises

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