Name: lass: ate: I: lgebra 2 Honors-hapter 6 Exam Short nswer 1. The base of a triangle is given by the expression 2x + 1. Its area is 2x 3 + 11x 2 + 9x + 2. Find a polynomial expression that represents its height. 2. The given equation E(x) = 2x 5 + 5x 4-3x 3 + 3x 2-5x + 1 shows the growth of a particular bacteria where x represents the number of days. Explain the process to solve for the number of days where the bacteria could have zero growth. 3. spacecraft has traveled 5 10 6 miles since it was launched. The length of the spacecraft is 3 10-4 miles. Find the quotient of the distance traveled divided by the length of the spacecraft. Factor completely. If the polynomial is not factorable, write prime. 4. x 6 + 8y 6 5. The perimeter of a triangle is 8x + 3y. The length of two sides of the triangle are 4x + y and 2x - 3y. Find the measure of the third side of the triangle. Essay 6. Ryan is making a circular pattern using rose petals. In his pattern, there is a petal at the center which is surrounded by 3 petals. Surrounding the first ring of petals, there is a second ring with 7 petals. The third ring has 15 petals, and so on. The total number of petals in his pattern can be modeled by the function p(n) = 4n 2-5n + 5 where n is the number of rings and p(n) is the number of petals in the pattern. Use the equation to find the number of petals in the eighth ring of the pattern. Give some examples of patterns existing in nature that can be modeled by a polynomial equation. 7. Explain why scientific notation is important in astronomy. The distance from Proxima entauri to the earth is 40,100,000,000,000 kilometers and the distance from the earth to the sun is 149,700,000 kilometers. Write both the distances in words and in scientific notation. Find the quotient of the distance from Proxima entauri to the earth divided by the distance from the earth to the sun. Write it in standard notation and in scientific notation. 1
Name: I: Multiple hoice Identify the choice that best completes the statement or answers the question. Simplify the given expression. ssume that no variable equals 0. 8. Ê 20x 20 y 9 ˆ ËÁ a. b. 40x 7 y 13 x 52 4 16y 16 c. x 52 y -16 16 d. x 13 16y 4 x 52 2y 16 Simplify the expression using long division. 9. (2x 2-33x + 16) (x - 16) a. quotient 2x - 1 and remainder 32 c. quotient 2x - 33 and remainder 16 b. quotient 2x + 1 and remainder 32 d. quotient 2x - 1 and remainder 0 2
Name: I: For the given graph, a. describe the end behavior, b. determine whether it represents an odd-degree or even-degree polynomial function, and c. state the number of real zeros. 10. a. The end behavior of the graph is f( x) Æ - as x Æ + and f( x) Æ + as x Æ -. It is an odd-degree polynomial function. The function has three real zeros. b. The end behavior of the graph is f( x) Æ + as x Æ + and f( x) Æ - as x Æ -. It is an odd-degree polynomial function. The function has three real zeros. c. The end behavior of the graph is f( x) Æ + as x Æ + and f( x) Æ + as x Æ -. It is an even-degree polynomial function. The function has three real zeros. d. The end behavior of the graph is f( x) Æ + as x Æ + and f( x) Æ + as x Æ -. It is an odd-degree polynomial function. The function has four real zeros. 3
Name: I: 11. a. The end behavior of the graph is f( x) Æ + as x Æ + and f( x) Æ + as x Æ -. It is an even-degree polynomial function. The function has five real zeros. b. The end behavior of the graph is f( x) Æ + as x Æ + and f( x) Æ + as x Æ -. It is an odd-degree polynomial function. The function has four real zeros. c. The end behavior of the graph is f( x) Æ + as x Æ + and f( x) Æ + as x Æ -. It is an even-degree polynomial function. The function has four real zeros. d. The end behavior of the graph is f( x) Æ + as x Æ + and f( x) Æ - as x Æ -. It is an even-degree polynomial function. The function has four real zeros. Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some of the factors may not be binomials. 12. 36x 3 + 60x 2-143x - 242; x - 2 a. (6x + 11)(6x - 11) c. (6x + 11)(6x + 11) b. 2(6x + 11) d. (6x - 11) 2 Factor the polynomial completely. 13. 30x 3-50x 2 + 27x - 45 a. (10x 2 + 9)(3x - 5) c. 10x 2 (3x - 5) - 27x + 45 b. (30x 3-50x 2 ) + (27x - 45) d. 10x 2 (3x - 5) - 9(3x - 5) 14. 10x 2-29x + 18 a. 10x 2-20x - 9x + 18 c. 10x 2-19x - 10x + 18 b. ( 10x - 9) ( x - 2) d. 10x(x - 2) - 9(x - 2) 4
Name: I: Simplify the given expression. 15. -2xy(3xy 3-5xy + 7y 2 ) a. -6x 2 y 4-5xy + 7y 2 c. -6x 2 y 4 + 10x 2 y 2-14xy 3 b. -6x 2 y 4-5x 2 y 2 + 7x 2 y 3 d. -6x 2 y 4 + 10xy + 14y 2 Ê 16. 11x 2 ˆ + 3x + 19 ËÁ + Ê 6x 2 ËÁ - 18x - 8 ˆ a. 29x 2-15x + 11 c. 17x 2 + 21x + 27 b. 17x 2 + 9x + 11 d. 17x 2-15x + 11 17. Find all of the zeros of the function f(x) = x 3-15x 2 + 73x - 111. a. 6 i, 6 + i c. 3, 6 i b. 3, 6 i, 6 + i d. 3, 6 i, 6 + i 18. List all of the possible rational zeros of the following function. f( x) = x 6-8x 5-21x 4 + 80x 3 + 20x 2-17x + 100 a. 1, 2, 4, 5, 10, 20, 25, 50 b. ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50 c. ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50, ±100 d. 1, 2, 4, 5, 10, 20, 25, 50, 100 Simplify the expression using synthetic division. 19. (6x 3-48x 2 + 120x - 96) (x - 4) a. Ê quotient 24x 2 ˆ + 48x + 312 ËÁ and remainder 1,152 b. Ê quotient 30x 2 ˆ + 72x - 408 ËÁ and remainder 1,536 c. Ê quotient 6x 2 ˆ - 72x - 168 ËÁ and remainder 576 d. Ê quotient 6x 2 ˆ - 24x + 24 ËÁ and remainder 0 20. Use synthetic substitution to find g ( 2) and g( 8) for the function g ( x) = x 5-7x 3-9x + 10. a. 48, 7,762 c. 80, 29,122 b. 4, 29,246 d. 32, 29,102 5
I: lgebra 2 Honors-hapter 6 Exam nswer Section SHORT NSWER 1. NS: 2x 2 + 10x + 4 rea,, of a triangle with base b and height h is given by the formula = 1 bh. Replace by the 2 expression 2x 3 + 11x 2 + 9x + 2 and b by the expression 2x + 1. Use division of polynomials to find the height of the triangle. PTS: 1 IF: dvanced REF: Lesson 6-2 OJ: 6-2.3 ivide polynomials using long and synthetic division. ST: M.912..4.4 TOP: Solve multi-step problems. KEY: Multistep Problems 2. NS: 4, 2 or 0; 1; 0, 2 or 4 If f(x) is a polynomial with real coefficients, the terms of which are arranged in descending powers of the variable, the number of positive real zeros of y = P(x) is the same as the number of changes in sign of the coefficients of the terms, or is less than by an even number, and the number of negative real zeros of y = P(x) is the same as the number of changes in sign of the coefficients of the terms of P( x), or is less than by an even number. Imaginary zeros comes in conjugate pairs, so the number of imaginary zeros must be an even number. PTS: 1 IF: verage REF: Lesson 6-7 OJ: 6-7.3 etermine the number and types of roots for and find the zeros of a polynomial equation. ST: M.912..4.6 M.912..4.8 M.912..4.3 M.912..4.7 TOP: Solve multi-step problems. KEY: Multistep Problems 3. NS: 1.7 10 10 mi Use the Properties of Powers to divide the numbers in scientific notation. PTS: 1 IF: asic REF: Lesson 6-1 OJ: 6-1.7 Use properties of exponents to multiply and divide monomials. Use expressions written in scientific notation and add, subtract and multiply monomials. ST: M.912..4.2 TOP: Solve multi-step problems. KEY: Multistep Problems 4. NS: (x 2 + 2y 2 ),(x 4 2x 2 y 2 + 4y 4 ) PTS: 1 1
I: 5. NS: 2x + 5y The measure of the third side of the triangle is given by the expression (8x + 3y) - (4x + y) - (2x - 3y). Simplify this expression by grouping similar terms and then combining them. PTS: 1 IF: verage REF: Lesson 6-1 OJ: 6-1.7 Use properties of exponents to multiply and divide monomials. Use expressions written in scientific notation and add, subtract and multiply monomials. ST: M.912..4.2 TOP: Solve multi-step problems. KEY: Multistep Problems 2
I: ESSY 6. NS: The number of petals in a pattern with 8 rings is given by p(8) = 4 8 2-5 8 + 5 = 221. The number of petals in a pattern with 7 rings is given by p(7) = 4 7 2-5 7 + 5 = 166. Thus, the number of petals in the eighth ring of the pattern is given by p(8) - p(7) = 221-166 = 55. Patterns found in nature that can be modeled by polynomial functions include spider webs and ripples formed when a stone is thrown in a pond. The number of petals in the eighth ring of the pattern is given by p(8) - p(7), where p(8) is the number of petals in a pattern with 8 rings and p(7) is the number of petals in a pattern with 7 rings. Spider webs can be modeled by a polynomial equation. ssessment Rubric Level 3 Superior *Shows thorough understanding of concepts. *Uses appropriate strategies. *omputation is correct. *Written explanation is exemplary. *iagram/table/chart is accurate (as applicable). *Goes beyond requirements of problem. Level 2 Satisfactory *Shows understanding of concepts. *Uses appropriate strategies. *omputation is mostly correct. *Written explanation is effective. *iagram/table/chart is mostly accurate (as applicable). *Satisfies all requirements of problem. Level 1 Nearly Satisfactory *Shows understanding of most concepts. *May not use appropriate strategies. *omputation is mostly correct. *Written explanation is satisfactory. *iagram/table/chart is mostly accurate (as applicable). *Satisfies most of the requirements of problem. Level 0 Unsatisfactory *Shows little or no understanding of the concept. *May not use appropriate strategies. *omputation is incorrect. *Written explanation is not satisfactory. *iagram/table/chart is not accurate (as applicable). *oes not satisfy requirements of problem. PTS: 1 IF: dvanced REF: Lesson 6-3 OJ: 6-3.3 Evaluate polynomial functions and identify general shapes of graphs of polynomial 3
I: functions. ST: M.912..4.5 TOP: Solve problems and show solutions. KEY: Problem Solving 4
I: 7. NS: stronomy often deals with very large numbers. For example, the approximate number of stars in the Milky Way Galaxy is 200,000,000,000. Such numbers, written in standard form, are difficult to work with because they contain too many digits. Scientific notation uses powers of 10 to make very large numbers more manageable. Thus, scientific notation is important in astronomy. The distance from the earth to Proxima entauri is forty trillion, one hundred billion kilometers or 4.01 10 13 km. The distance to the sun from the earth is one hundred forty-nine million, seven hundred thousand kilometers or 1.497 10 8 13 4.01 10 km. The quotient obtained is 1.497 10 ª 2.679 10 5 km or 8 267,900 km. Scientific notation is useful in astronomy because it makes it easier to write the large numbers involved. The scientific notation of a number is given by a 10 n, where 1 a < 10 and n is an integer. Use the Properties of Powers to divide numbers in scientific notation. ssessment Rubric Level 3 Superior *Shows thorough understanding of concepts. *Uses appropriate strategies. *omputation is correct. *Written explanation is exemplary. *iagram/table/chart is accurate (as applicable). *Goes beyond requirements of problem. Level 2 Satisfactory *Shows understanding of concepts. *Uses appropriate strategies. *omputation is mostly correct. *Written explanation is effective. *iagram/table/chart is mostly accurate (as applicable). *Satisfies all requirements of problem. Level 1 Nearly Satisfactory *Shows understanding of most concepts. *May not use appropriate strategies. *omputation is mostly correct. *Written explanation is satisfactory. *iagram/table/chart is mostly accurate (as applicable). *Satisfies most of the requirements of problem. Level 0 Unsatisfactory *Shows little or no understanding of the concept. *May not use appropriate strategies. *omputation is incorrect. *Written explanation is not satisfactory. *iagram/table/chart is not accurate (as applicable). *oes not satisfy requirements of problem. 5
I: PTS: 1 IF: dvanced REF: Lesson 6-1 OJ: 6-1.7 Use properties of exponents to multiply and divide monomials. Use expressions written in scientific notation and add, subtract and multiply monomials. ST: M.912..4.2 TOP: Solve problems and show solutions. KEY: Problem Solving MULTIPLE HOIE 8. NS: Simplify each base using the properties of powers. Then, write all the fractions in the simplest terms and ensure there are no negative exponents. orrect! There should be no negative exponents. Raise the numerator and the denominator to the fourth power before simplifying. Use the Power of a Power Property to all the terms in the monomial. PTS: 1 IF: verage REF: Lesson 6-1 OJ: 6-1.2 Use properties of exponents to divide monomials. ST: M.912..4.2 TOP: Use properties of exponents to divide monomials. KEY: Monomials ivide Monomials 9. NS: Use the division algorithm. When dividing, you can add or subtract only similar terms. hange the signs of the product terms only. id you use the correct signs of the terms? id you consider both the terms of the divisor? orrect! PTS: 1 IF: dvanced REF: Lesson 6-2 OJ: 6-2.1 ivide polynomials using long division. TOP: ivide polynomials using long division. KEY: Polynomials ivide Polynomials Long ivision ST: M.912..4.4 6
I: 10. NS: The end behavior is the behavior of the graph as x approaches positive infinity ( + ) or negative infinity (- ). The x-coordinate of the point at which the graph intersects the x-axis is called the zero of the function. orrect! What is the end behavior of the graph? heck the degree of the polynomial function. id you verify the number of real zeros? PTS: 1 IF: asic REF: Lesson 6-3 OJ: 6-3.2 Identify general shapes of graphs of polynomial functions. ST: M.912..4.5 TOP: Identify general shapes of graphs of polynomial functions. KEY: Polynomial Functions Graph Polynomial Functions 11. NS: The end behavior is the behavior of the graph as x approaches positive infinity ( + ) or negative infinity (- ). The x-coordinate of the point at which the graph intersects the x-axis is called the zero of the function. id you verify the number of real zeros? heck the degree of the polynomial function. orrect! What is the end behavior of the graph? PTS: 1 IF: asic REF: Lesson 6-3 OJ: 6-3.2 Identify general shapes of graphs of polynomial functions. ST: M.912..4.5 TOP: Identify general shapes of graphs of polynomial functions. KEY: Polynomial Functions Graph Polynomial Functions 12. NS: Use the Factor Theorem. id you verify the answer by multiplying the factors? id you factor correctly? orrect! Is the square of the binomial equal to the depressed polynomial? PTS: 1 IF: dvanced REF: Lesson 6-6 OJ: 6-6.2 etermine whether a binomial is a factor of a polynomial by using synthetic substitution. ST: M.912..4.6 M.912..4.8 M.912..4.3 TOP: etermine whether a binomial is a factor of a polynomial by using synthetic substitution. KEY: Polynomial Functions Synthetic Substitution 7
I: 13. NS: Group the monomials to find the GF (greatest common factor), factor the GF of each binomial, and then use the istributive Property to obtain the factors. orrect! Factor the GF of each binomial. Group the polynomial into binomials to find the GF. Use the istributive Property. PTS: 1 IF: verage REF: Lesson 6-5 OJ: 6-5.2 Factor polynomials by grouping. ST: M.912..4.3 TOP: Factor polynomials by grouping. KEY: Polynomials Factor Polynomials 14. NS: To find the coefficient of the x terms, find two numbers whose product is 10 18 or 180 and whose sum is 29. Factor the GF of each group. orrect! The product of the coefficient of the x terms should be equal to the product of the coefficient of the x 2 term and the constant term. Use the istributive Property to obtain two binomial factors. PTS: 1 IF: verage REF: Lesson 6-5 OJ: 6-5.3 Factor polynomials with addition recognizing the FOIL method. ST: M.912..4.3 TOP: Factor polynomials with addition by recognizing the FOIL method. KEY: Polynomials Factor Polynomials FOIL Method 15. NS: Use the istributive Property and then multiply the monomials using the Product of Powers Property. id you use the istributive Property? id you multiply the monomials correctly? orrect! id you calculate the product of powers correctly? PTS: 1 IF: verage REF: Lesson 6-1 OJ: 6-1.6 Multiply polynomials. ST: M.912..4.2 TOP: Multiply polynomials. KEY: Polynomials Multiply Polynomials 8
I: 16. NS: Group similar terms and then combine them. id you group the similar terms? id you group the similar terms and then combine them? id you combine the similar terms correctly? orrect! PTS: 1 IF: verage REF: Lesson 6-1 OJ: 6-1.4 dd polynomials. ST: M.912..4.2 TOP: dd polynomials. KEY: Polynomials dd Polynomials 17. NS: Use synthetic substitution to obtain the required answer. The function also has a positive real zero. There is no change in sign for the coefficients of f( x). Find all the possible imaginary zeros. orrect! PTS: 1 IF: verage REF: Lesson 6-7 OJ: 6-7.2 Find the zeros of a polynomial function. ST: M.912..4.6 M.912..4.8 M.912..4.3 M.912..4.7 TOP: Find the zeros of a polynomial function. KEY: Polynomial Functions Zeroes of Polynomial Functions 18. NS: Use the Rational Zero Theorem. id you consider the negative rational zeros? id you calculate all the zeros correctly? orrect! You must also include the positive rational zeros in the answer. PTS: 1 IF: verage REF: Lesson 6-8 OJ: 6-8.1 Identify the possible rational zeros of a polynomial function. ST: M.912..4.6 M.912..4.8 M.912..4.3 TOP: Identify the possible rational zeros of a polynomial function. KEY: Polynomial Functions Zeroes of Polynomial Functions 9
I: 19. NS: To use synthetic division, the divisor must be of the form x - r. Multiply the first coefficient with the constant in the divisor and bring it below the second coefficient. ring the first coefficient below itself in the third row. dd the product of the constant in the divisor to the coefficient above it. orrect! PTS: 1 IF: dvanced REF: Lesson 6-2 OJ: 6-2.2 ivide polynomials using synthetic division. TOP: ivide polynomials using synthetic division. KEY: Polynomials ivide Polynomials Synthetic ivision 20. NS: Use synthetic substitution to obtain the required answer. ST: M.912..4.4 The degree of the function is 5, not 4. id you calculate correctly? id you substitute the correct values? orrect! PTS: 1 IF: dvanced REF: Lesson 6-6 OJ: 6-6.1 Evaluate functions using synthetic substitution. ST: M.912..4.6 M.912..4.8 M.912..4.3 TOP: Evaluate functions using synthetic substitution. KEY: Polynomial Functions Synthetic Substitution 10