9.11 Complex Rationals

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1 Unit 9 ~ Contents Algebra Beaut and Awe ~ Newton Dividing Larger Polnomials Polnomial Division With a Binomial Remainder Quadratic Equations:Taking the Square Root of Both Sides Dividing Radicals Adding and Subtracting Rationals With Unlike Denominators Quiz How Attractive! Permutations and Probabilit Rationalizing Denominators Quadratic Equations: Completing the Square Dividing Polnomials With Missing Terms Quiz In Perfect Balance Comple Rationals Quadratic Equations: Solving b Completing the Square The Domain of a Function Rational Equations Review for Test Test Under the Arch

2 9. Comple Rationals A comple rational is a fraction over another fraction. Below are several eamples. Figure Figure Figure Figure 4 a b + a b + Two methods can be used to simplif comple rationals. The best method to use depends on the nature of the epression. Method. This method works best with a single fraction over another single fraction, such as in Figures and above. Because a fraction bar indicates division, rewrite the epression horizontall using the division sign. Multipl b the reciprocal to complete the division. Eample. Division changed to multiplication b the reciprocal. Division rewritten horizontall. 4 Simplified answer. Eample. a b a a a a b ab b b b a a b b ab 0 ~ Algebra I Unit 9

3 Eample. 9 Division changed to multiplication b the reciprocal. 9 9 ( + )( ) ( + ) Division rewritten horizontall. Simplified answer. Method. This method works best when the numerator and/or denominator of a comple rational contains more than one term, as is the case in Figures and 4 on the previous page. To simplif such epressions, find the LCD of all the terms in the numerator and denominator. Then eliminate the fractions b multipling the numerator and the denominator b the LCD. Eample 4. ( ) ( ) (0) 8 0 (0) is the LCD of the three terms,, and. Multipling the numerator and the denominator b 0 eliminates the fractions. Eample. + + ( + ) ( + ) () + + () ( + ) ( + ) is the LCD of the four terms,,, and. Multipling the numerator and the denominator b eliminates the fractions. 9. Comple Rationals ~

4 Eample. + ( ) ( + ) () ( + )( ) + 8 ( + ) () is the LCD of the four terms,,, and. Multipling the numerator and the denominator b eliminates the fractions. Simplif z 9 z m n 4. p.. + mn p Think of as. + Review Divide ( + 8) ( + ). ( + 4) ( ) Complete the square in the quadratic equations ~ Algebra I Unit 9

5 Factor polnomials completel. 8.. k k + 8k Follow the directions The Famil Friendl lunch buffet has choices of meat, 8 choices of vegetables, choices of fruits/salads, and 7 choices of desserts. If a temperate customer took onl one of each choice, how man possible varieties of meal are there? 9. Nine women are in the Good Samaritan Sewing Circle. How man permutations eist for choosing a President, Secretar, and Treasurer? What is the probabilit that Caleb s mother will be Secretar, Kezia s mother will be President, and Guillermo s mother will be Treasurer? Round the answer to the nearest hundredth of a percent. Factor the trinomials. If an is not a perfect square, write not a perfect square Divide the rational epressions ( 9) Solve the sstems of equations, using the method of our choice..4,., Simplif Comple Rationals ~

6 9. Quadratic Equations: Solving b Completing the Square Lesson 9.8 stated that completing the square could be used to solve an quadratic equation. After completing the square, three more steps are needed to solve quadratic equations. First, the perfect square trinomial created b completing the square is factored into a binomial squared. Then the square root of both sides of the equation is taken. Finall, the value(s) for are found b solving the remaining equation. Eample. Solve the quadratic equation Original equation. Constant moved to the right side. 49 added to both sides to complete the square. ( + 7) Perfect square trinomial factored. ( + 7) ± Square root taken of both sides. 7 ± 7 subtracted from both sides. The solutions are: and. Eample. Solve the quadratic equation Original equation. 0 Both sides divided b the coefficient of the term () Simplified. Constant term moved to the right side. 9 added to both sides to complete the square. ( + ) 4 Perfect square trinomial factored. ( + ) ± 4 Square root taken of both sides. ± 4 subtracted from both sides. The solutions are: + 4 and 4. 4 ~ Algebra I Unit 9

7 Eample. Complete the square of the quadratic equation Original equation. + 0 Both sides divided b the coefficient of the term (). + 0 Simplified. + Constant term moved to the right side added to both sides to complete the square. ( + ) 49 4 Perfect square trinomial factored. 7 + ± 4 4 Square root taken of both sides. 7 ± subtracted from both sides The solutions are and. Complete the square and solve the quadratic equations Review Simplif. 9. r s s r + Solve the quadratic equations for b taking the square root of both sides. 9.. ( + ) Quadratic Equations: Solving b Completing the Square ~

8 Simplif b rationalizing denominators r s Simplif b dividing radicals Divide ( ) ( + ) 4. ( + ) ( + ) Solve b using direct variation If n varies directl as m, and n is when m is 7, what is n when m is?. If varies directl as the square of, and is. when is, what is when is? 7. The distance awa ou are from a hunter is directl proportional to the time it takes before ou hear the sound of the rifle. If someone 0 meters awa hears the shot in. seconds, how far awa is someone who hears it in onl second after the shot? 8. Think about the last problem. What is the constant of proportionalit and what does it represent? Solve the quadratic equations b factoring Graph these sstems of inequalities , and 8 40., and, and Simplif the epressions Add or subtract. Leave our answers in factored form e e + (e + ) 9. ( ) 4 ( + 4) ~ Algebra I Unit 9

9 Use a sstem of equations to solve the problems An isoceles triangle whose base is half the length of a side has a perimeter of. centimeters. How long are the sides? 4. Adam drove 80 more miles than Jared on an,00 mile trip. How man miles did each drive? Solve b using inverse variation If s varies inversel as the square of r, and s is 0.00 when r is 0, what is s when r is 0? 4. An empt space shuttle orbiter weighs,0 pounds at sea level. How much less does it weigh at the edge of the stratosphere, miles above the earth? (Remember that weight is inversel proportional to the square of the distance from the center of the earth which is approimatel 4,000 miles in radius.) 44. If n varies inversel as m, and n is 90 when m is, what is n when m is? Solve for the requested information Gene began to divide the profits from his business into two savings accounts: 7% for his personal needs, and the other % to an account to be donated to a mission hospital in 8 months when it is scheduled to begin. Both accounts earn the same interest rate of %. At the end of 8 months the total interest for the two accounts was $70. What were the principals in each account (these would in realit be the average principal amounts over 8 months, but treat them as lump sum investments)? (Hint: The principal in the personal account was times the principal in the mission account.) Complete the square and solve the quadratic equations Quadratic Equations: Solving b Completing the Square ~ 7

10 9. The Domain of a Function One wa to think of functions is as machines which accept input value and produce output values. The input value is called the independent variable () and the output value is called the dependent variable (). The function tells what will happen with the input to produce the output. input () 4 function output () + The set of possible input values () is the domain of the function. The domain can be unlimited such as the set of all real numbers, as is the case for the function above: +. Then an positive or negative real number can be used for. At other times the domain has limits such as the function in which cannot be 0: domain {real numbers, 0}. The domain of a function is usuall determined from a given list of ordered pairs, an equation, or its graph. When a list of ordered pairs is given, the domain is the -values of the ordered pairs. If an equation or a graph is given, the domain is all the values of the -ais that the graph covers. Eample. Give the domain for each of the following. a. (, 7) (0, 8) (, 9) (4, 7) (, ) domain {, 0,, 4, } b. 7 domain {all real numbers} c. domain {real numbers, } d. + 4 domain {all real numbers} e. domain {real numbers, 0} 8 ~ Algebra I Unit 9

11 Eample. Give the domain for each graph. a. b. c. domain {real numbers, 0} domain {real numbers, < } domain {real numbers} Specif the domain and tell whether or not it represents a function. 9.. (, 8) (, 7) (4, 4) (, ) The Domain of a Function ~ 9

12 7. 8. Review Complete the square and solve the quadratic equations Combine like radicals. If the cannot be combined, write cannot be combined Divide. 9.. (8h + 8h + h ) (4h + h + 8). ( 8 + ) ( ) Add or subtract as indicated. 8.4 j + j j + j j j Simplif Factor b grouping r + 4rs 7r 7s ~ Algebra I Unit 9

13 Specif the domain and tell whether or not it represents a function The Domain of a Function ~ 4

Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1

Math 030 Review for Final Exam Revised Fall 2010 RH/ DM 1 Math 00 Review for Final Eam Revised Fall 010 RH/ DM 1 1. Solve the equations: (-1) (7) (-) (-1) () 1 1 1 1 f. 1 g. h. 1 11 i. 9. Solve the following equations for the given variable: 1 Solve for. D ab