Problem Set 10: Polynomials
Practice Makes Perfect
Identify Polynomials, Monomials, Binomials and Trinomials In the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial. binomial trinomial polynomial monomial Determine the Degree of Polynomials In the following exercises, determine the degree of each polynomial. 5 1 0 Add and Subtract Monomials In the following exercises, add or subtract the monomials. 15x2 −8u 5a + 7b Add: −4a −3b Add: 16x Subtract −17x6 Subtract Add and Subtract Polynomials In the following exercises, add or subtract the polynomials. 12y2 + 4y + 8 −3x2 + 17x − 1 4a2 − 7a − 11 4m2 − 10m + 2 11z + 8 12s2 − 16s + 9 Find the sum of and 2p3 + p2 + 9p + 10 Find the sum of and Subtract from x2 + 3x + 4 Subtract from Find the difference of and 11w − 66 Find the difference of and Evaluate a Polynomial for a Given Value In the following exercises, evaluate each polynomial for the given value. ⓐ ⓑ ⓒ ⓐ 187 ⓑ 40 ⓒ 2 ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ ⓐ −104 ⓑ 4 ⓒ 40 ⓐ ⓑ ⓒ A window washer drops a squeegee from a platform feet high. The polynomial gives the height of the squeegee seconds after it was dropped. Find the height after seconds. 19 feet A manufacturer of microwave ovens has found that the revenue received from selling microwaves at a cost of p dollars each is given by the polynomial . Find the revenue received when dollars.Everyday Math
Fuel Efficiency The fuel efficiency (in miles per gallon) of a bus going at a speed of miles per hour is given by the polynomial . Find the fuel efficiency when 10 mpg Stopping Distance The number of feet it takes for a car traveling at miles per hour to stop on dry, level concrete is given by the polynomial . Find the stopping distance whenWriting Exercises
Using your own words, explain the difference between a monomial, a binomial, and a trinomial. Answers will vary. Eloise thinks the sum is . What is wrong with her reasoning?Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ If most of your checks were: …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific. …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? …no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.Practice Makes Perfect
Simplify Expressions with Exponents In the following exercises, simplify each expression with exponents. 1,024 0.008 625 −625 −10,000 −.25 Simplify Expressions Using the Product Property of Exponents In the following exercises, simplify each expression using the Product Property of Exponents. x9 a5 314 z6 xa+2 ya+b Simplify Expressions Using the Power Property of Exponents In the following exercises, simplify each expression using the Power Property of Exponents. u8 y20 1012 x90 x2y 5xy Simplify Expressions Using the Product to a Power Property In the following exercises, simplify each expression using the Product to a Power Property. 25a2 −216m3 16r2s2 256x4y4z4 Simplify Expressions by Applying Several Properties In the following exercises, simplify each expression. x14 a36 45x3 200a5 8m18 1,000x6y3 16a12b8 1,024a10 25,000p24 x18y18 144m8n22 Multiply Monomials In the following exercises, multiply the following monomials. −60x6 72u7 4r11 36a5b7 8x2y5Everyday Math
Email Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is , on the third round is , as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.Round | Number of people |
---|---|
Year | Salary |
---|---|
\text{$40,000}\left(1.05\right) | |
\text{$40,000}{\left(1.05\right)}^{2} | |
\text{$40,000}{\left(1.05\right)}^{3} | |
Writing Exercises
Use the Product Property for Exponents to explain why . Answers will vary. Explain why but . Jorge thinks is . What is wrong with his reasoning? Answers will vary. Explain why is , and not .Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ After reviewing this checklist, what will you do to become confident for all objectives?Practice Makes Perfect
Multiply a Polynomial by a Monomial In the following exercises, multiply. 4x + 40 15r − 360 −3m − 33 −8z + 40 u2 + 5u n3 − 3n2 12x2 − 120x −27a2 − 45a 24x2 + 6xy 55p2 − 25pq 3v2 + 30v + 75 8n3 − 8n2 + 2n −8y3 − 16y2 + 120y 5q5 − 10q4 + 30q3 −12z4 − 48z3 + 4z2 2y2 − 9y 8w − 48 Multiply a Binomial by a Binomial In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical method x2 + 10x + 24 n2 + 9n − 36 In the following exercises, multiply the following binomials. Use any method. y2 + 11y + 24 a2 + 22a + 96 u2 − 14u + 45 z2 − 32z + 220 x2 + 3x − 28 v2 + 7v − 60 6n2 + 11n + 5 20m2 − 88m − 9 16c2 − 1 15u2 − 82u + 112 2a2 + 5ab + 3b2 5x2 − 20x − xy + 4y Multiply a Trinomial by a Binomial In the following exercises, multiply using ⓐ the Distributive Property and ⓑ the Vertical Method. u3 + 7u2 + 14u + 8 3a3 + 31a2 + 5a − 50 In the following exercises, multiply. Use either method. y3 − 16y2 + 69y − 54 2x3 − 9x2 − 17x − 6Everyday Math
Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply times . Think of as and as .- ⓐ Multiply by the FOIL method.
- ⓑ Multiply without using a calculator.
- ⓒ Which way is easier for you? Why?
- ⓐ 195
- ⓑ 195
- ⓐ Answers will vary.
- ⓐ Multiply by the FOIL method.
- ⓑ Multiply without using a calculator.
- ⓒ Which way is easier for you? Why?
Writing Exercises
Which method do you prefer to use when multiplying two binomials—the Distributive Property, the FOIL method, or the Vertical Method? Why? Answers will vary. Which method do you prefer to use when multiplying a trinomial by a binomial—the Distributive Property or the Vertical Method? Why?Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?Practice Makes Perfect
Simplify Expressions Using the Quotient Property of Exponents In the following exercises, simplify. 46 x9 r4 Simplify Expressions with Zero Exponents In the following exercises, simplify. 1 1 −1 ⓐ ⓑ ⓐ 1 ⓑ 10 ⓐ ⓑ ⓐ ⓑ ⓐ 1 ⓑ −27x5 ⓐ ⓑ ⓐ ⓑ ⓐ 1 ⓑ 15 ⓐ ⓑ 7 Simplify Expressions Using the Quotient to a Power Property In the following exercises, simplify. Simplify Expressions by Applying Several Properties In the following exercises, simplify. x3 u2 1 a14 y3 1 3x8 Divide Monomials In the following exercises, divide the monomials. 8b6 2x 5u4v3Mixed Practice
ⓐ ⓑ ⓒ ⓓ ⓐ ⓑ ⓒ ⓓ ⓐ ⓑ ⓒ ⓓ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑEveryday Math
Memory One megabyte is approximately bytes. One gigabyte is approximately bytes. How many megabytes are in one gigabyte? Memory One megabyte is approximately bytes. One terabyte is approximately bytes. How many megabytes are in one terabyte? 1,000,000Writing Exercises
Vic thinks the quotient simplifies to . What is wrong with his reasoning? Mai simplifies the quotient by writing . What is wrong with her reasoning? Answers will vary. When Dimple simplified and she got the same answer. Explain how using the Order of Operations correctly gives different answers. Roxie thinks simplifies to . What would you say to convince Roxie she is wrong? Answers will vary.Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?Practice Makes Perfect
Use the Definition of a Negative Exponent In the following exercises, simplify. ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ ⓐ ⓑ ⓒ Simplify Expressions with Integer Exponents In the following exercises, simplify. r3 m6 b8 m7 p3 Convert from Decimal Notation to Scientific Notation In the following exercises, write each number in scientific notation. 45,000 280,000 2.8 × 105 8,750,000 1,290,000 1.29 × 106 0.036 0.041 4.1 × 10−2 0.00000924 0.0000103 1.03 × 10−5 The population of the United States on July 4, 2010 was almost . The population of the world on July 4, 2010 was more than . 6.85 × 109 The average width of a human hair is centimeters. The probability of winning the Megamillions lottery is about . 5.7 × 10−9 Convert Scientific Notation to Decimal Form In the following exercises, convert each number to decimal form. 830 16,000,000,000 0.028 0.0000000615 In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was . At the start of 2012, the US federal budget had a deficit of more than \text{$1.5}\times {10}^{13}. $15,000,000,000,000 The concentration of carbon dioxide in the atmosphere is . The width of a proton is of the width of an atom. 0.00001 Multiply and Divide Using Scientific Notation In the following exercises, multiply or divide and write your answer in decimal form. 0.003 0.00000735 200,000 50,000,000Everyday Math
Calories In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by trillion calories by the end of 2015.- ⓐ Write trillion in decimal notation.
- ⓑ Write trillion in scientific notation.
- ⓐ Write this number in scientific notation.
- ⓑ How many years does it take for the difference to become 1 day?
- ⓐ 1.25 × 10−4
- ⓐ 8,000
Writing Exercises
- ⓐ Explain the meaning of the exponent in the expression .
- ⓑ Explain the meaning of the exponent in the expression
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?Practice Makes Perfect
Find the Greatest Common Factor of Two or More Expressions In the following exercises, find the greatest common factor. 15 25 4 5 3x 12p3 2a 10y 5x3 7b2 Factor the Greatest Common Factor from a Polynomial In the following exercises, factor the greatest common factor from each polynomial. 5(y + 3) 4(b − 5) 7(x − 1) 3(n2 + 7n + 4) 6(q2 + 5q + 7) c(9c + 22) x(17x + 7) q(4q + 7) 3r(r + 9) 10u(3u − 1) b(a + 8) 11y(5 − y3) 15c2(3c − 1) 6c(c2 − d2) 24x2(2x + 3) 18a3(8a3 + 5) 10(y2 + 5y + 4) 12(u2 − 3u − 9) 5p2(p2 − 4p − 3) 8c3(c2 + 5c − 7) −7(p + 12) −6b(3b + 11) −8a2(a − 4) −9b3(b2 − 7)Everyday Math
Revenue A manufacturer of microwave ovens has found that the revenue received from selling microwaves a cost of dollars each is given by the polynomial . Factor the greatest common factor from this polynomial. Height of a baseball The height of a baseball hit with velocity feet/second at feet above ground level is , with the number of seconds since it was hit. Factor the greatest common factor from this polynomial. −4(4t2 − 20t − 1)Writing Exercises
The greatest common factor of and is . Explain what this means. What is the GCF of , , and ? Write a general rule that tells how to find the GCF of , , and .Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?Chapter Review Exercises
Add and Subtract Polynomials
Identify Polynomials, Monomials, Binomials and Trinomials In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial. trinomial binomial Determine the Degree of Polynomials In the following exercises, determine the degree of each polynomial. 2 0 Add and Subtract Monomials In the following exercises, add or subtract the monomials. 15p Add −3n5 Subtract from Add and Subtract Polynomials In the following exercises, add or subtract the polynomials. 10a2 + 4a − 1 6y2 − 3y + 3 Find the sum of and 8q3 + q2 + 6q − 29 Find the difference of and Evaluate a Polynomial for a Given Value of the Variable In the following exercises, evaluate each polynomial for the given value. 995 2,955 −163 A pair of glasses is dropped off a bridge feet above a river. The polynomial gives the height of the glasses seconds after they were dropped. Find the height of the glasses when . 64 feet The fuel efficiency (in miles per gallon) of a bus going at a speed of miles per hour is given by the polynomial . Find the fuel efficiency when mph.Use Multiplication Properties of Exponents
Simplify Expressions with Exponents In the following exercises, simplify. 216 0.25 Simplify Expressions Using the Product Property of Exponents In the following exercises, simplify each expression. p13 a6 Simplify Expressions Using the Power Property of Exponents In the following exercises, simplify each expression. y12 310 Simplify Expressions Using the Product to a Power Property In the following exercises, simplify each expression. 64n2 256a8b8 Simplify Expressions by Applying Several Properties In the following exercises, simplify each expression. 27a15 x21 Multiply Monomials In the following exercises, multiply the monomials. −54p5 56x3y11Multiply Polynomials
Multiply a Polynomial by a Monomial In the following exercises, multiply. 70 − 7x −625y4 + 5y Multiply a Binomial by a Binomial In the following exercises, multiply the binomials using various methods. a2 + 7a + 10 6x2 − 19x − 7 n2 + 9n + 8 5u2 + 37u − 24 p2 + 11p + 28 27c2 − 3c − 4 Multiply a Trinomial by a Binomial In the following exercises, multiply using any method. x3 − 2x2 − 24x − 21 m3 − m2 − 72m − 180Divide Monomials
Simplify Expressions Using the Quotient Property of Exponents In the following exercises, simplify. 26or 64 Simplify Expressions with Zero Exponents In the following exercises, simplify. 1 1 Simplify Expressions Using the Quotient to a Power Property In the following exercises, simplify. Simplify Expressions by Applying Several Properties In the following exercises, simplify. a2 Divide Monomials In the following exercises, divide the monomials. 9p9Integer Exponents and Scientific Notation
Use the Definition of a Negative Exponent In the following exercises, simplify. Simplify Expressions with Integer Exponents In the following exercises, simplify. x6 k6 b10 Convert from Decimal Notation to Scientific Notation In the following exercises, write each number in scientific notation. 5.3 × 106 The thickness of a piece of paper is about millimeter. 9.7 × 10−2 According to www.cleanair.com, U.S. businesses use about tons of paper per year. Convert Scientific Notation to Decimal Form In the following exercises, convert each number to decimal form. 29,000 375 Multiply and Divide Using Scientific Notation In the following exercises, multiply and write your answer in decimal form. 6,000 30,000,000,000Introduction to Factoring Polynomials
Find the Greatest Common Factor of Two or More Expressions In the following exercises, find the greatest common factor. 5 4x2 Factor the Greatest Common Factor from a Polynomial In the following exercises, factor the greatest common factor from each polynomial. 8(2u − 3) 6p(p + 1) −9a3(a2 + 1) 5(y2 − 11y + 9)Chapter Practice Test
For the polynomial- ⓐ Is it a monomial, binomial, or trinomial?
- ⓑ What is its degree?
- ⓐ trinomial
- ⓑ 4