Rational and Radical Expressions and Equations
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1 Rational and Radical Epressions and Equations Secondary Mathematics Page 44 Jordan School District
2 Unit Cluster 7 (AAPR6 and AAPR7): Rational Epressions Cluster 7: Rewrite rational epressions 7 Rewrite simple rational epressions in different forms using inspection, long division, or, for the more complicated eamples, a computer algebra system (CAS) 7 Add, subtract, multiply, and divide rational epressions 7 Closure of rational epressions under addition, subtraction, multiplication, and division by a nonzero rational epression VOCABULARY A rational function is a function of the form polynomials and q 0 For eample, f f p q, where p and 4 is a rational function 1 q are Eample 1: Simplify f f 4 6 f f 4 6 Simplify 0 because the denominator must be a nonzero polynomial Rewrite the rational epression as the sum of fractions with a common denominator Eample : Simplify f 4 4 f f f f because the denominator must be a nonzero polynomial Factor Simplify like terms Secondary Mathematics Page 4 Jordan School District
3 Eample : Simplify f 4 f f f f Factor or 1 because the denominator must be a nonzero polynomial Simplify like terms Eample 4: Simplify f 81 1 f f f f 81 1 Factor because the denominator must be a nonzero polynomial Simplify like terms Practice Eercises A Simplify each rational epression Secondary Mathematics Page 46 Jordan School District
4 Long Division with Polynomials A rational epression f p can be thought of as q p divided by polynomials, like division of real numbers, uses multiplication and subtraction q Division of Let us review the algorithm for long division with real numbers Consider divided by is the dividend is the divisor Rewrite the division problem so that the dividend is under the long division symbol and the divisor is on the outside Divide by is a little over one This becomes the first term of your quotient Multiply by 1 and subtract the product from the dividend Bring down the net unused term This is your new dividend Divide 10 by 10 is a little over four This becomes the net term of your quotient Multiply by 4 and subtract the product from the dividend The remainder is 1 Therefore, 1 14 Secondary Mathematics Page 47 Jordan School District
5 Eample : Simplify using long division is the dividend 4 is the divisor Rewrite the rational epression with the dividend under the long division symbol and the divisor on the outside Divide by this becomes the first term of your quotient Multiply and subtract the product from the dividend Bring down the net unused term This is your new dividend Divide by quotient Multiply from the dividend The remainder is zero Therefore, by this becomes the net term of your 4 by and subtract the product Eample 6: Simplify using long division is the dividend is the divisor Rewrite the rational epression with the dividend under the long division symbol and the divisor on the outside Divide by of your quotient Multiply by the product from the dividend this becomes the first term and subtract Bring down the net unused term This is your new dividend Secondary Mathematics Page 48 Jordan School District
6 Divide by this becomes the net term of your quotient Multiply by and subtract the product from the dividend Bring down the net unused term This is your new dividend Divide 6 by 6 this becomes the net term of your quotient Multiply by and subtract the product from the dividend The remainder is Therefore, 8 7 Practice Eercises B Simplify using long division Secondary Mathematics Page 49 Jordan School District
7 Using Technology to Divide Polynomials For more complicated polynomial division you may want to use a computer algebra system such as the TI-Nspire CAS You can download an app of the TI-Nspire CAS for your ipad or you can purchase a TI-Nspire CAS calculator The instructions below are for the ipad app version Eample 7: Divide 4 8 by Create a new document by pushing the + symbol in the upper left hand corner A menu, like the one at the right, will appear Select Calculator You will have a document that you can type mathematical equations in Push the wrench at the top of the screen to bring up the Tools menu then select Algebra A new Algebra menu will appear Scroll down and Select Polynomial Tools The Polynomial Tools menu should have Remainder of Polynomial and Quotient of Polynomial You will need both of these tools to divide Select Quotient of Polynomial Enter the dividend polynomial (for this eample it is 4 8 ) then a comma and the divisor polynomial (for this eample it is ) Press Enter and you will have the quotient, but not the remainder if there is one Secondary Mathematics Page 0 Jordan School District
8 Push the wrench to bring up the Polynomial Tools menu again Select Remainder of Polynomial Enter the dividend polynomial 4 (for this eample it is 8 ) then a comma and the divisor polynomial (for this eample it is ) Press Enter and you will have the remainder 4 8 divided by is You can also use Geogebra as a computer algebra system The program is free and can be downloaded to your computer at wwwgeogebraorg (Instructions for PC version) Eample 7B: Divide 4 8 by When you open Geogebra you should see a menu like the one at the right Select CAS & Graphics If it there is no menu, then press ctrl + shift + k to bring up the CAS screen You should see a screen like the one at the right To close the Graphics screen click on the in the right hand corner To the right of the number 1, start typing division and some options will come up Select Division [<Dividend Polynomial>, <Divisor Polynomial>] The dividend polynomial is 4 8 and the divisor polynomial is Once you have entered the polynomials, press enter and you will get an answer in the form {Quotient,Remainder} 4 8 divided by is Secondary Mathematics Page 1 Jordan School District
9 Practice Eercises C Use a computer algebra system to divide the polynomials Multiplying and Dividing Rational Epressions VOCABULARY If the numerator and denominator of a rational epression have no common factors, other than then the rational epression is in simplified form 1, To multiply a rational epression by another, multiply the numerator with the numerator and the denominator with the denominator a c ac Simplify if possible ( b 0 and d 0 ) b d bd ac bd To divide one rational epression by another, multiply the first epression by the reciprocal of the second epression a c a d ad Simplify b d b c bc ad bc if possible ( b 0, d 0, and c 0) Eample 8: y 8 z Multiply z 1y y 8 z z 1y 4 0z y yz 4 0 y z 1 y z Separate like terms Multiply the numerator with the numerator and the denominator with the denominator Secondary Mathematics Page Jordan School District
10 y z Simplify using integer eponent properties 4 yz Multiply the numerator with the numerator and the denominator with the denominator Eample 9: 1 4 Multiply Factor Identify the like factors Simplify the like factors Multiply the numerator with the numerator and the denominator with the denominator Eample 10: 4 Multiply Factor Identify the like factors Simplify the like factors Multiply the numerator with the numerator and the denominator with the denominator Secondary Mathematics Page Jordan School District
11 Eample 11: Divide yz 1y z 7y 7 y z yz 1y z 7y 7 y z yz 7 y z 7y 1y z y z 10 y z y z Separate like terms y z y 1 y Multiply by the reciprocal of the term that follows the division symbol Multiply the numerator with the numerator and the denominator with the denominator Simplify using integer eponent properties 10 Multiply the numerator with the numerator and the denominator with the denominator Eample 1: 10 9 Divide Factor Identify the like factors Simplify the like factors Multiply the numerator with the numerator and the denominator with the denominator Secondary Mathematics Page 4 Jordan School District
12 Practice Eercises D Perform the indicated operation, if possible, simplify Determine if your answer is a rational epression 1 8 y 9y 4 9y y 6 7 y 4 y y 4 y z z 6y y Simplify YOU DECIDE Are rational epressions closed under multiplication and division Justify your conclusion using the method of your choice Secondary Mathematics Page Jordan School District
13 Adding and Subtracting Rational Epressions VOCABULARY To add a rational epression to another, find a common denominator then add the numerators a c a d c b ad cb ad cb Simplify b d b d d b bd bd bd d 0 ) ad cb bd if possible ( b 0 and To subtract one rational epression from another, find a common denominator then subtract the numerators a c a d c b ad cb ad cb Simplify b d b d d b bd bd bd d 0 ) ad cb bd if possible ( b 0 and The least common multiple (LCM) for epressions is the smallest (non-zero) epression that is a multiple of two or more epressions Let us review how to add rational numbers Add 1 6 and You need to find a common denominator The least common multiple of 6 and 1 is 0 This will be the common denominator so multiply Add the numerators Simplify 1 6 by 0 1 so multiply 1 by Adding and subtracting rational epressions is similar to adding and subtracting rational numbers The key is finding the common denominator or the least common multiple of the denominators Eample 1: Find the least common multiple for the epressions a 10 y and 1y b 1 and 1 c 11 4 and 1 6 Secondary Mathematics Page 6 Jordan School District
14 a 10 y and 1y 10 y y 1y 10 y y y y y 1 y y y y y 6y 60 y Find the prime factorizations of each epression Identify what factors the epressions have in common Then identify what is unique to both epressions and 6y y Multiply the common factors and the unique factors to obtain the least common multiple b 1 and Factor each epression Identify what factors the epressions have in common Then identify what is unique to both epressions 1 1 Multiply the common factors and the unique factors to obtain the least common multiple c 11 4 and Factor each epression Identify what factors the epressions have in common 8 Then identify what is unique to both epressions and Multiply the common factors and the unique factors to obtain the least common multiple Eample 14: Perform the indicated operation If possible, simplify a 4 b c 4 a Add the numerators Simplify Secondary Mathematics Page 7 Jordan School District
15 b Subtract the numerators Simplify c Subtract the numerators Simplify Eample 1: Perform the indicated operations If possible, simplify a b c a Factor and determine the LCM The LCM is epression by epression by 1 1 Add the numerators Simplify Multiply the first 1 Multiply the second Secondary Mathematics Page 8 Jordan School District
16 b There are no common factors so the LCM is epression by 1 epression by Multiply the first 1 Multiply the second 1 1 Subtract the numerators Simplify c Factor and determine the LCM The LCM is first epression by second epression by 1 Subtract the numerators Simplify Multiply the 1 Multiply the Secondary Mathematics Page 9 Jordan School District
17 Practice Eercises E Perform the indicated operation, if possible, simplify Determine if your answer is a rational epression Simplify y 6y 1 1 YOU DECIDE Are rational epressions closed under addition and subtraction Justify your conclusion using the method of your choice Secondary Mathematics Page 60 Jordan School District
18 Unit Cluster 8 (AREI): One Variable Rational and Radical Equations Cluster 8: Understand solving equations as a process of reasoning and eplain the reasoning 8 Solve simple rational equations in one variable and give eamples of how etraneous solutions may arise 8 Solve simple radical equations in one variable and give eamples of how etraneous solutions may arise VOCABULARY A rational equation is an equation that contains one or more rational epressions (ie, 1 is a rational equation) 4 An etraneous solution is a solution of an equation that has been transformed or derived from the original equation but it is not a solution of the original equation When working with rational functions you must check the solution in the original equation Finding Restrictions on Rational Equations The denominator of a rational epression cannot be zero When solving a rational equation, any values that would make any denominator zero must be ecluded as possible answers These are referred to as restrictions Eample 1: Find the restrictions for each rational equation a b a The possible answers cannot include Set the denominator equal to zero and solve for 4 b Factor the denominator of the epression on the right side Secondary Mathematics Page 61 Jordan School District
19 The possible answers cannot include 8 and Set each unique factor equal to zero and solve for Notice that the factor is repeated 1 8 Practice Eercises A Find the restrictions for each rational equation Solving Rational Equations To solve a rational equation: determine any values that would make the denominator zero find a common denominator by finding the least common multiple of the denominators multiply all of the terms on both sides of the equation by the common denominator to eliminate the fractions simplify and solve the resulting equation compare your answer with the restrictions to ensure that it is valid Secondary Mathematics Page 6 Jordan School District
20 Eample : Solve Determine any numbers that will make the denominator zero The answer cannot be 0 The common denominator is 6 Multiply each term by 6 and simplify Simplify and solve the equation Eample : Solve Factor the denominators Find any values that will make the denominators zero The answer cannot be or and The common denominator is Multiply each term by simplify Simplify and solve the new equation Compare the answer against the restrictions to make sure that it is valid Secondary Mathematics Page 6 Jordan School District
21 Eample 4: Find any values that will make the denominators zero The answer cannot be 1 and 1 or The common denominator is 1 Multiply each term by simplify Simplify and solve the new equation Compare the answer against the restrictions to make sure that it is valid Eample : Solve Find any values that will make the denominators zero The answer cannot be 1 The common denominator is 1 Multiply each term by 1 and simplify Simplify and solve the equation 1 The mathematical answer is 1, but this value will make the denominator zero, therefore, 1 is an etraneous solution and there is no solution to this equation Secondary Mathematics Page 64 Jordan School District
22 Practice Eercises B Solve each equation Secondary Mathematics Page 6 Jordan School District
23 VOCABULARY A radical equation is an equation that has a variable in a radicand or a 1/ variable with a rational eponent (ie, 4 or (4 1) 1) The radicand is the epression under the radical sign The inde is the small number outside of the radical sign Solving Radical Equations To solve a radical equation: isolate the radical on one side of the equation raise each side to the power of the inde simplify check solutions in the original equation to eliminate any etraneous solutions Eample 6: Solve Isolate the radical term by subtracting 4 from each side Square each side of the equation Solve for Check the solution in the original equation Secondary Mathematics Page 66 Jordan School District
24 Eample 7: Solve Isolate the radical term by adding 1 to each side Cube each side of the equation Solve for Check the solution in the original equation Eample 8: Solve Isolate the radical term by adding 7 to each side Square each side of the equation Remember that Solve for Secondary Mathematics Page 67 Jordan School District
25 The only solution is etraneous solution because Check the solutions in the original equation 8 does not work in the original equation so it is an Eample 9: Solve Isolate one of the radical terms Square each side of the equation Remember that Isolate the radical term Solve for Square each side of the equation Remember that Check the solutions in the original equation The only solution is 8 because 0 does not work in the original equation so it is an etraneous solution Secondary Mathematics Page 68 Jordan School District
26 Practice Eercises C Solve each radical equation HONORS Recall that when a rational number is multiplied by its reciprocal the product is 1 (ie, a a b ab b 1) To solve radical equations of the form k, raise each side of the equation to b a ab the power of the reciprocal Eample 10: Solve 1 / 1 b a 1 / 1 / 1 8 / 1 4 Isolate the radical term Secondary Mathematics Page 69 Jordan School District
27 / / / / / / / The reciprocal of of the equation to the is Raise each side power Rewrite the using the properties of eponents Remember that the square root of a number has a positive and a negative solution Check the solutions in the original equation Both 7 and 9 are solutions to the radical equation Eample 11: Solve / 8 / / / Isolate the radical term Rewrite the epression in rational eponent form The reciprocal of is Raise each side of the equation to the power Rewrite the using the properties of eponents Remember that the cube root has only one real answer Secondary Mathematics Page 70 Jordan School District
28 , Check the solution in the original equation 0 is a solution to the radical equation Practice Eercises D Solve each radical equation 1 7 / / / / / 9 4/ Secondary Mathematics Page 71 Jordan School District
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