ALGEBRA 2 FINAL EXAM REVIEW
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1 Class: Date: ALGEBRA 2 FINAL EXAM REVIEW Multiple Choice Identify the choice that best completes the statement or answers the question.. Classify 6x 5 + x + x 2 + by degree. quintic c. quartic cubic d. quadratic 2. Classify 8x + 7x + 5x by number of terms. trinomial c. polynomial of 5 terms binomial d. polynomial of terms Consider the leading term of each polynomial function. What is the end behavior of the graph?. 5x 8! 2x 7! 8x 6 + The leading term is 5x 8. Since n is even and a is positive, the end behavior is down and up. The leading term is 5x 8. Since n is even and a is positive, the end behavior is up and down. c. The leading term is 5x 8. Since n is even and a is positive, the end behavior is up and up. d. The leading term is 5x 8. Since n is even and a is positive, the end behavior is down and down..!x 5 + 9x + 5x + The leading term is!x 5. Since n is odd and a is negative, the end behavior is up and up. The leading term is!x 5. Since n is odd and a is negative, the end behavior is down and down. c. The leading term is!x 5. Since n is odd and a is negative, the end behavior is up and down. d. The leading term is!x 5. Since n is odd and a is negative, the end behavior is down and up. Write the polynomial in factored form. 5. x + 9x 2 + 8x 6x(x + )(x + ) c. x(x + 6)(x ) x(x + 6)(x + ) d. x(x + )(x + 6)
2 What are the zeros of the function? Graph the function. 6. y = x(x! 2)(x + 5) 2, 5 c. 0, 2, 5 0, 2, 5 d. 2, 5, 2 7. Divide x + 2x 2 + x + by x +. x 2! x + 59 c. x 2! x + 59, R 22 x 2 + 8x! 5, R 20 d. x 2 + 8x! 5 Divide using synthetic division. 8. Divide!6x + 8x 2! 7x! 0 by (x! 2).!6x 2 + 6x + 5 c.!6x 2 + 0x! 9, R 20 6x 2! 6x! 5 d. 6x 2! 0x + 9, R 20 2
3 9. Use the Rational Root Theorem to list all possible rational roots of the polynomial equation x! 6x 2 + x + 9 = 0. Do not find the actual roots. 9,,, 9 c. 9,,,,, 9,, 9 d. no roots 0. (d! 2) 6 Use Pascal s Triangle to expand the binomial. d 6 + 2d d + 60d + 20d d d 6! 6d 5 + 5d! 20d + 5d 2! 6d + c. d 6! 2d d! 60d + 20d 2! 92d + 6 d. d 6 + 6d 5 + 5d + 20d + 5d 2 + 6d +. Find all the real square roots of and c and and d and 0.02 Find the real-number root. 2.! ! 25 c.! d.! 5 7 Multiply and simplify if possible.. " c. Ê ˆ. 7x x! 7 7 ËÁ x 7! 9 x c. x 7! x 9 7x! 9x d.! 2x d. What is the simplest form of the expression? 5. 08a 6 b 9 a 5 b a a 5 b a c. a 5 b a d. none of these
4 What is the simplest form of the product? 6. 50x 7 y 7 " 6xy 2x y 6 75y c. 5x y 6 2 0x y 5 y d. 0x y 5 y What is the simplest form of the quotient? c. d. What is the simplest form of the radical expression? 8. 2a! 6 2a!6 2a c.! 2a 9 2a d. not possible to simplify What is the simplest form of the expression? ! 5 5 c d. 5 5 What is the product of the radical expression? Ê ˆ Ê 20. 7! 2 ËÁ ËÁ ˆ c ! 2 d Ê ˆ Ê 5! 2 ËÁ ËÁ ˆ 2 c d. 8
5 How can you write the expression with rationalized denominator? 22.! 6 + 6!! 2 8 c.! + 2 2!! d. 9! c. d Simplify. 2. " 9 9 c. d c. 6 2 d Write the exponential expression x in radical form. 8 x 8 x c. x 8 d. 8 8 x What is the solution of the equation? 27. 2x + 8! 6 =! 2 c. 2 d. 28. ( x + 6) 5 = 8 2 c. 26 d. 8 5
6 29. Let f(x) = x! 5 and g(x) = 6x!. Find f(x)! g(x). 0x 8 0x 2 c. 2x 8 d. 2x 2 0. Let f(x) = x + 2 and g(x) = 7x + 6. Find f " g and its domain. 6x 2 + x + 2; all real numbers except x =! 2 6x 2 + x + 2; all real numbers c. 2x 2 + 2x + 2; all real numbers d. 2x 2 + 2x + 2; all real numbers except x =! 6 7. Let f(x) = x 2! 6 and g(x) = x +. Find f g and its domain. x + ; all real numbers except x # x + ; all real numbers except x #! c. x! ; all real numbers except x # d. x! ; all real numbers except x #! 2. Let f(x) = x + 2 and g(x) = x 2. Find Ê ËÁ g û fˆ (!5). 9 c. 9 d. 0 What is the inverse of the given relation?. y = 7x 2!. y = ± x + 7 x = y + 7 c. y 2 = x! 7 d. y = ± x! 7. y = x + 9 y = x + c. y = x + y = x! d. y = x! 6
7 5. y = x Graph the exponential function. c. d. 6. Suppose you invest $600 at an annual interest rate of.6% compounded continuously. How much will you have in the account after years? $ $6,70.28 c. $0,8.07 d. $, How much money invested at 5% compounded continuously for years will yield $820? $ $88.8 c. $780.0 d. $ Write the equation in logarithmic form = 2 log 2 = 5 " 2 c. log 2 = 5 log 2 2 = 5 d. log 5 2 = 2 7
8 Evaluate the logarithm. 9. log c. d. 0. log c. d.. log c. 2 d. 0 Write the expression as a single logarithm. 2. log b q + 6 log b v log b (q v 6 ) c. ( + 6) log Ê b ËÁ q + vˆ Ê log b qv + 6 ˆ ËÁ d. log Ê q + v 6 ˆ b ËÁ. log! log 2 log 2 log 2 c. log 2 d. log 2. log Expand the logarithmic expression. d 2 5. log p log d! log 2 c. log d log 2!d log 2 d. log 2! log d log " log p c. log + log p log! log p d. log p 6. Use the Change of Base Formula to evaluate log c d..7 8
9 Solve the exponential equation = 6 x! 2 c. 7 2 d x = 8 8 c. 8 d. 2 Solve the logarithmic equation. Round to the nearest ten-thousandth if necessary. 9. log 2x = c d Find the horizontal asymptote of the graph of y =!x 6 + 6x + 8x 6 + 9x +. y = c. y = 0 y =! 2 d. no horizontal asymptote Simplify the rational expression. State any restrictions on the variable p 2! p! 2 p +!p + 8; p #! c.!p! 8; p # p! 8; p #! d. p + 8; p # q 2 + q + 2 q 2! 5q! 2 q + 8 q + 8 q! 8 ; q #!, q #!8 c.!(q + 8) q! 8 ; q # 8 d.!(q + 8) q! 8 ; q #!, q # 8 q! 8 ; q #!, q # 8 9
10 What is the product in simplest form? State any restrictions on the variable. 5. a 5 7b " a 9 2b 2 2a 7b 6, a # 0, b # 0 c. a 7 b a 2, a # 0, b # 0 7 b 2, a # 0, b # 0 d. 7 a 9 b 6, a # 0, b # 0 What is the quotient in simplified form? State any restrictions on the variable. 5. x + 2 x + x! x 2 + x! 5 (x + 2)(x + 5), x #! 5,! c. x + (x + 2)(x + ) (x! ) 2 (x + 5) (x + 2)(x + ) (x! ) 2 (x + 5), x #,! 5 d. (x + 2)(x + 5) x +, x #,! 5,!, x #,! Simplify the sum a a 2! 6 7a! 9 (a! 8)(a + 8) a 2 + a! 56 c. d. (a! 8)(a + 8) 7a + 6 (a! 8)(a + 8) Simplify the difference. 56. b 2! 2b! 8 6 b 2 + b! 2! b! b! 0 c. b 2! 2b! b 2 + b! 2 d. b! b! b! 0 b! 0
11 Simplify the complex fraction t! t 2t + 2t! 5! c.! 5 d.! 58. x + x + 2x + x 2 + x x x + 9 Solve the equation. Check the solution. x c. x 2 + 0x + d. not here 59.!2 x + x + =! 6! c.! 8 d.! 60. a 2 a 2! 6 + a! 6 = a c. 9 and 6 d. 6
12 ALGEBRA 2 FINAL EXAM REVIEW Answer Section MULTIPLE CHOICE. ANS: A PTS: DIF: L2 REF: 5- Polynomial Functions OBJ: 5-. To classify polynomials STA: MA.92.A.2.5 MA.92.A..5 TOP: 5- Problem Classifying Polynomials KEY: degree of a polynomial polynomial function standard form of a polynomial function DOK: DOK 2. ANS: D PTS: DIF: L2 REF: 5- Polynomial Functions OBJ: 5-. To classify polynomials STA: MA.92.A.2.5 MA.92.A..5 TOP: 5- Problem Classifying Polynomials KEY: degree of a polynomial polynomial function standard form of a polynomial function DOK: DOK. ANS: C PTS: DIF: L2 REF: 5- Polynomial Functions OBJ: 5-.2 To graph polynomial functions and describe end behavior STA: MA.92.A.2.5 MA.92.A..5 TOP: 5- Problem 2 Describing End Behavior of Polynomial Functions KEY: polynomial end behavior DOK: DOK. ANS: C PTS: DIF: L REF: 5- Polynomial Functions OBJ: 5-.2 To graph polynomial functions and describe end behavior STA: MA.92.A.2.5 MA.92.A..5 TOP: 5- Problem 2 Describing End Behavior of Polynomial Functions KEY: polynomial end behavior DOK: DOK 5. ANS: D PTS: DIF: L2 REF: 5-2 Polynomials, Linear Factors, and Zeros OBJ: 5-2. To analyze the factored form of a polynomial STA: MA.92.A.. MA.92.A..5 MA.92.A..7 MA.92.A..8 TOP: 5-2 Problem Writing a Polynomial in Factored Form KEY: 6. ANS: C PTS: DIF: L REF: 5-2 Polynomials, Linear Factors, and Zeros OBJ: 5-2. To analyze the factored form of a polynomial STA: MA.92.A.. MA.92.A..5 MA.92.A..7 MA.92.A..8 TOP: 5-2 Problem 2 Finding Zeros of a Polynomial Function 7. ANS: C PTS: DIF: L2 REF: 5- Dividing Polynomials OBJ: 5-. To divide polynomials using long division STA: MA.92.A.. MA.92.A.. MA.92.A..6 TOP: 5- Problem Using Polynomial Long Division KEY: 8. ANS: A PTS: DIF: L REF: 5- Dividing Polynomials OBJ: 5-.2 To divide polynomials using synthetic division STA: MA.92.A.. MA.92.A.. MA.92.A..6 TOP: 5- Problem Using Synthetic Division KEY: synthetic division
13 9. ANS: C PTS: DIF: L2 REF: 5-5 Theorems About Roots of Polynomial Equations OBJ: 5-5. To solve equations using the Rational Root Theorem STA: MA.92.A..6 MA.92.A..7 TOP: 5-5 Problem Finding a Rational Root KEY: Rational Root Theorem DOK: DOK 0. ANS: C PTS: DIF: L2 REF: 5-7 The Binomial Theorem OBJ: 5-7. To expand a binomial using Pascal's Triangle STA: MA.92.A..2 TOP: 5-7 Problem Using Pascal's Triangle KEY: Pascal's Triangle expand. ANS: D PTS: DIF: L REF: 6- Roots and Radical Expressions OBJ: 6-. To find nth roots STA: MA.92.A.0. TOP: 6- Problem Finding All Real Roots KEY: nth root DOK: DOK 2. ANS: D PTS: DIF: L REF: 6- Roots and Radical Expressions OBJ: 6-. To find nth roots STA: MA.92.A.0. TOP: 6- Problem 2 Finding Roots KEY: radicand index nth root DOK: DOK. ANS: D PTS: DIF: L2 REF: 6-2 Multiplying and Dividing Radical Expressions OBJ: 6-2. To multiply and divide radical expressions STA: MA.92.A.6.2 MA.92.A.0. TOP: 6-2 Problem Multiplying Radical Expressions KEY: DOK: DOK. ANS: A PTS: DIF: L REF: 6-2 Multiplying and Dividing Radical Expressions OBJ: 6-2. To multiply and divide radical expressions STA: MA.92.A.6.2 MA.92.A.0. TOP: 6-2 Problem Multiplying Radical Expressions KEY: 5. ANS: A PTS: DIF: L REF: 6-2 Multiplying and Dividing Radical Expressions OBJ: 6-2. To multiply and divide radical expressions STA: MA.92.A.6.2 MA.92.A.0. TOP: 6-2 Problem 2 Simplifying a Radical Expression KEY: simplest form of a radical DOK: DOK 6. ANS: B PTS: DIF: L REF: 6-2 Multiplying and Dividing Radical Expressions OBJ: 6-2. To multiply and divide radical expressions STA: MA.92.A.6.2 MA.92.A.0. TOP: 6-2 Problem Simplifying a Product KEY: simplest form of a radical 7. ANS: A PTS: DIF: L2 REF: 6-2 Multiplying and Dividing Radical Expressions OBJ: 6-2. To multiply and divide radical expressions STA: MA.92.A.6.2 MA.92.A.0. TOP: 6-2 Problem Dividing Radical Expressions KEY: simplest form of a radical DOK: DOK 8. ANS: C PTS: DIF: L2 REF: 6- Binomial Radical Expressions OBJ: 6-. To add and subtract radical expressions STA: MA.92.A.6.2 TOP: 6- Problem Adding and Subtracting Radical Expressions KEY: like radicals DOK: DOK 2
14 9. ANS: A PTS: DIF: L REF: 6- Binomial Radical Expressions OBJ: 6-. To add and subtract radical expressions STA: MA.92.A.6.2 TOP: 6- Problem Simplifying Before Adding or Subtracting 20. ANS: B PTS: DIF: L2 REF: 6- Binomial Radical Expressions OBJ: 6-. To add and subtract radical expressions STA: MA.92.A.6.2 TOP: 6- Problem Multiplying Binomial Radical Expressions DOK: DOK 2. ANS: A PTS: DIF: L REF: 6- Binomial Radical Expressions OBJ: 6-. To add and subtract radical expressions STA: MA.92.A.6.2 TOP: 6- Problem 5 Multiplying Conjugates DOK: DOK 22. ANS: C PTS: DIF: L REF: 6- Binomial Radical Expressions OBJ: 6-. To add and subtract radical expressions STA: MA.92.A.6.2 TOP: 6- Problem 6 Rationalizing the Denominator DOK: DOK 2. ANS: D PTS: DIF: L2 REF: 6- Binomial Radical Expressions OBJ: 6-. To add and subtract radical expressions STA: MA.92.A.6.2 TOP: 6- Problem 6 Rationalizing the Denominator DOK: DOK 2. ANS: C PTS: DIF: L REF: 6- Rational Exponents OBJ: 6-. To simplify expressions with rational exponents STA: MA.92.A.6. MA.92.A.6. TOP: 6- Problem Simplifying Expressions with Rational Exponents KEY: rational exponents DOK: DOK 25. ANS: B PTS: DIF: L2 REF: 6- Rational Exponents OBJ: 6-. To simplify expressions with rational exponents STA: MA.92.A.6. MA.92.A.6. TOP: 6- Problem Simplifying Expressions with Rational Exponents KEY: rational exponents DOK: DOK 26. ANS: A PTS: DIF: L2 REF: 6- Rational Exponents OBJ: 6-. To simplify expressions with rational exponents STA: MA.92.A.6. MA.92.A.6. TOP: 6- Problem 2 Converting Between Exponential and Radical Form KEY: rational exponents DOK: DOK 27. ANS: B PTS: DIF: L2 REF: 6-5 Solving Square Root and Other Radical Equations OBJ: 6-5. To solve square root and other radical equations STA: MA.92.A.6. MA.92.A.6.5 MA.92.A.0. TOP: 6-5 Problem Solving a Square Root Equation KEY: square root equation 28. ANS: C PTS: DIF: L REF: 6-5 Solving Square Root and Other Radical Equations OBJ: 6-5. To solve square root and other radical equations STA: MA.92.A.6. MA.92.A.6.5 MA.92.A.0. TOP: 6-5 Problem 2 Solving Other Radical Equations KEY: radical equation 29. ANS: D PTS: DIF: L REF: 6-6 Function Operations OBJ: 6-6. To add, subtract, multiply, and divide functions STA: MA.92.A.2.7 MA.92.A.2.8 TOP: 6-6 Problem Adding and Subtracting Functions 0. ANS: C PTS: DIF: L REF: 6-6 Function Operations OBJ: 6-6. To add, subtract, multiply, and divide functions STA: MA.92.A.2.7 MA.92.A.2.8 TOP: 6-6 Problem 2 Multiplying and Dividing Functions
15 . ANS: D PTS: DIF: L REF: 6-6 Function Operations OBJ: 6-6. To add, subtract, multiply, and divide functions STA: MA.92.A.2.7 MA.92.A.2.8 TOP: 6-6 Problem 2 Multiplying and Dividing Functions 2. ANS: A PTS: DIF: L REF: 6-6 Function Operations OBJ: To find the composite of two functions STA: MA.92.A.2.7 MA.92.A.2.8 TOP: 6-6 Problem Composing Functions KEY: composite function. ANS: A PTS: DIF: L REF: 6-7 Inverse Relations and Functions OBJ: 6-7. To find the inverse of a relation or function STA: MA.92.A.2. TOP: 6-7 Problem 2 Finding an Equation for the Inverse KEY: inverse relation. ANS: D PTS: DIF: L REF: 6-7 Inverse Relations and Functions OBJ: 6-7. To find the inverse of a relation or function STA: MA.92.A.2. TOP: 6-7 Problem 2 Finding an Equation for the Inverse KEY: inverse relation 5. ANS: D PTS: DIF: L2 REF: 7- Exploring Exponential Models OBJ: 7-. To model exponential growth and decay STA: MA.92.A.8. MA.92.A.8. MA.92.A.8.7 TOP: 7- Problem Graphing an Exponential Function KEY: exponential function 6. ANS: D PTS: DIF: L2 REF: 7-2 Properties of Exponential Functions OBJ: To graph exponential functions that have base e STA: MA.92.A.2.5 MA.92.A.2.0 MA.92.A.8. MA.92.A.8. MA.92.A.8.7 TOP: 7-2 Problem 5 Continuously Compounded Interest KEY: continuously compounded interest 7. ANS: D PTS: DIF: L REF: 7-2 Properties of Exponential Functions OBJ: To graph exponential functions that have base e STA: MA.92.A.2.5 MA.92.A.2.0 MA.92.A.8. MA.92.A.8. MA.92.A.8.7 TOP: 7-2 Problem 5 Continuously Compounded Interest KEY: continuously compounded interest 8. ANS: B PTS: DIF: L2 REF: 7- Logarithmic Functions as Inverses OBJ: 7-. To write and evaluate logarithmic expressions STA: MA.92.A.2.5 MA.92.A.2. MA.92.A.8. MA.92.A.8. TOP: 7- Problem Writing Exponential Equations in Logarithmic Form KEY: logarithm 9. ANS: C PTS: DIF: L REF: 7- Logarithmic Functions as Inverses OBJ: 7-. To write and evaluate logarithmic expressions STA: MA.92.A.2.5 MA.92.A.2. MA.92.A.8. MA.92.A.8. TOP: 7- Problem 2 Evaluating a Logarithm KEY: logarithm
16 0. ANS: A PTS: DIF: L2 REF: 7- Logarithmic Functions as Inverses OBJ: 7-. To write and evaluate logarithmic expressions STA: MA.92.A.2.5 MA.92.A.2. MA.92.A.8. MA.92.A.8. TOP: 7- Problem 2 Evaluating a Logarithm KEY: logarithm. ANS: B PTS: DIF: L REF: 7- Logarithmic Functions as Inverses OBJ: 7-. To write and evaluate logarithmic expressions STA: MA.92.A.2.5 MA.92.A.2. MA.92.A.8. MA.92.A.8. TOP: 7- Problem 2 Evaluating a Logarithm KEY: logarithm 2. ANS: A PTS: DIF: L REF: 7- Properties of Logarithms OBJ: 7-. To use the properties of logarithms STA: MA.92.A.8.2 MA.92.A.8.6 TOP: 7- Problem Simplifying Logarithms. ANS: A PTS: DIF: L2 REF: 7- Properties of Logarithms OBJ: 7-. To use the properties of logarithms STA: MA.92.A.8.2 MA.92.A.8.6 TOP: 7- Problem Simplifying Logarithms. ANS: A PTS: DIF: L2 REF: 7- Properties of Logarithms OBJ: 7-. To use the properties of logarithms STA: MA.92.A.8.2 MA.92.A.8.6 TOP: 7- Problem 2 Expanding Logarithms 5. ANS: C PTS: DIF: L REF: 7- Properties of Logarithms OBJ: 7-. To use the properties of logarithms STA: MA.92.A.8.2 MA.92.A.8.6 TOP: 7- Problem 2 Expanding Logarithms 6. ANS: A PTS: DIF: L REF: 7- Properties of Logarithms OBJ: 7-. To use the properties of logarithms STA: MA.92.A.8.2 MA.92.A.8.6 TOP: 7- Problem Using the Change of Base Formula KEY: Change of Base Formula 7. ANS: C PTS: DIF: L REF: 7-5 Exponential and Logarithmic Equations OBJ: 7-5. To solve exponential and logarithmic equations STA: MA.92.A.8.5 TOP: 7-5 Problem Solving an Exponential Equation Common Base KEY: exponential equation 8. ANS: C PTS: DIF: L2 REF: 7-5 Exponential and Logarithmic Equations OBJ: 7-5. To solve exponential and logarithmic equations STA: MA.92.A.8.5 TOP: 7-5 Problem Solving an Exponential Equation Common Base KEY: exponential equation 9. ANS: A PTS: DIF: L2 REF: 7-5 Exponential and Logarithmic Equations OBJ: 7-5. To solve exponential and logarithmic equations STA: MA.92.A.8.5 TOP: 7-5 Problem 5 Solving a Logarithmic Equation KEY: logarithmic equation 50. ANS: B PTS: DIF: L REF: 8- Rational Functions and Their Graphs OBJ: 8-. To identify properties of rational functions STA: MA.92.A.5.6 TOP: 8- Problem Finding Horizontal Asymptotes KEY: rational function 5
17 5. ANS: B PTS: DIF: L2 REF: 8- Rational Expressions OBJ: 8-. To simplify rational expressions STA: MA.92.A.0. TOP: 8- Problem Simplifying a Rational Expression KEY: rational expression simplest form 52. ANS: C PTS: DIF: L REF: 8- Rational Expressions OBJ: 8-. To simplify rational expressions STA: MA.92.A.0. TOP: 8- Problem Simplifying a Rational Expression KEY: rational expression simplest form 5. ANS: B PTS: DIF: L2 REF: 8- Rational Expressions OBJ: 8-.2 To multiply and divide rational expressions STA: MA.92.A.0. TOP: 8- Problem 2 Multiplying Rational Expressions KEY: rational expression simplest form 5. ANS: D PTS: DIF: L REF: 8- Rational Expressions OBJ: 8-.2 To multiply and divide rational expressions STA: MA.92.A.0. TOP: 8- Problem Dividing Rational Expressions KEY: rational expression simplest form 55. ANS: A PTS: DIF: L2 REF: 8-5 Adding and Subtracting Rational Expressions OBJ: 8-5. To add and subtract rational expressions TOP: 8-5 Problem 2 Adding Rational Expressions 56. ANS: D PTS: DIF: L REF: 8-5 Adding and Subtracting Rational Expressions OBJ: 8-5. To add and subtract rational expressions TOP: 8-5 Problem Subtracting Rational Expressions 57. ANS: A PTS: DIF: L2 REF: 8-5 Adding and Subtracting Rational Expressions OBJ: 8-5. To add and subtract rational expressions TOP: 8-5 Problem Simplifying a Complex Fraction KEY: complex fraction 58. ANS: C PTS: DIF: L REF: 8-5 Adding and Subtracting Rational Expressions OBJ: 8-5. To add and subtract rational expressions TOP: 8-5 Problem Simplifying a Complex Fraction KEY: complex fraction 59. ANS: D PTS: DIF: L2 REF: 8-6 Solving Rational Equations OBJ: 8-6. To solve rational equations TOP: 8-6 Problem Solving a Rational Equation KEY: rational equation 60. ANS: A PTS: DIF: L REF: 8-6 Solving Rational Equations OBJ: 8-6. To solve rational equations TOP: 8-6 Problem Solving a Rational Equation KEY: rational equation 6
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