17.1 Understanding Polynomial Expressions

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1 COMMON CORE 4 a b Locker x LESSON Common Core Math Standards The student is expected to: COMMON CORE A-SSE.A.a Interpret parts of an expression, such as terms, factors, and coefficients. Also A-SSE.A.b, A-SSE.A., A-APR.A., A-CED.A. Mathematical Practices COMMON CORE 7. Understanding Polynomial Expressions MP. Reasoning Language Objective Explain to a partner how to find the degree of a polynomial. Name Class Date 7. Understanding Polynomial Expressions Essential Question: What are polynomial expressions, and how do you simplify them? Explore Identifying Monomials Resource Locker A monomial is an expression consisting of a number, variable, or product of numbers and variables that have whole number exponents. Terms of an expression are parts of the expression separated by plus signs. (Remember that x y can be written as x + ( y).) A monomial cannot have more than one term, and it cannot have a variable in its denominator. Here are some examples of monomials and expressions that are not monomials. Monomials 4 x -4xy 0.5 x Use the following process to determine if 5 ab is a monomial. xy_ 4 Not Monomials 4 + x x x x - y_ x ENGAGE Essential Question: What are polynomial expressions, and how do you simplify them? A polynomial expression is a monomial or a sum of monomials. You simplify them by combining like terms. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the designs left in the sky by aerial fireworks and the paths the fireworks follow before they explode. Then preview the Lesson Performance Task. A 5 ab has one term(s), so it could be a monomial. B Does 5 ab have a denominator? No C D E If possible, split it into a product of numbers and variables. 5 ab = 5 a b List the numbers and variables in the product. Numbers: 5 Variables: a, b Check the exponent of each variable. Complete the following table. Variable Exponent F The exponents of the variables in 5 ab are all whole numbers. Therefore, 5 ab is a monomial. G Is _ 5 k a monomial? No, a monomial cannot have a variable in the denominator. a b Module Lesson Name Class Date 7. Understanding Polynomial Expressions Essential Question: What are polynomial expressions, and how do you simplify them? A-SSE.A.a Interpret parts of an expression, such as terms, factors, and coefficients. Also A-SSE.A.b, A-SSE.A., A-APR.A., A-CED.A. Explore Identifying Monomials Monomials Not Monomials xy_ Resource A monomial is an expression consisting of a number, variable, or product of numbers and variables that have whole number exponents. Terms of an expression are parts of the expression separated by plus signs. (Remember that x y can be written as x + ( y).) A monomial cannot have more than one term, and it cannot have a variable in its denominator. Here are some examples of monomials and expressions that are not monomials. 4 x -4xy 0.5 x 4 + x x x - 0.5x - Use the following process to determine if 5 ab is a monomial. 5 ab has term(s), so it be a monomial. Does 5 ab have a denominator? No If possible, split it into a product of numbers and variables. 5 ab = 5 List the numbers and variables in the product. Numbers: Variables: Check the exponent of each variable. Complete the following table. Variable Exponent The exponents of the variables in 5 ab are all. one could ab 5 is a, b Therefore, 5 ab a monomial. Is 5_ k a monomial? whole numbers No, a monomial cannot have a variable in the denominator. Module Lesson y_ HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. 805 Lesson 7.

2 H Complete the table below. Term Is this a monomial? 5ab yes 5ab is the product of a number, 5, and the variables a and b. _ Identifying Monomials x is the product of the variable x, two times. _ y can be written as y. Variables must have whole number exponents. yes x EXPLORE Explain your reasoning. QUESTIONING STRATEGIES y no yes is equal to the number 4. It is the product of, two times. 5 _ k no A monomial cannot have a variable in the denominator. 5x + 7 no A monomial cannot have more than one term. x + 4ab no A monomial cannot have more than one term. Why is 0.x - not considered to be a monomial? In a monomial, a variable must have whole number exponents. Why is _ not considered to be a monomial? x Explain. A monomial cannot have a variable x - in the denominator. Also, = ; the exponent in x a monomial must be a whole number. k _ is the product of the variable k, two 4 times, times _. k _ 4 yes INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP. Make sure that students are clear on the 4 Reflect. Discussion Explain why 6 is a monomial but x is not a monomial. 6 is a monomial because it is a number, and numbers are monomials. x is not a definition of a monomial before they begin work on the problems. A monomial is a number, a variable, or a product of numbers and variables that have whole number exponents. monomial because monomials can only have variables with whole number exponents. Discussion Is x 0 a monomial? Justify your answer in two ways. Yes. x 0 equals, and is a number. Also, a variable with a whole number exponent is a monomial and 0 is a whole number. Explain Classifying Polynomials A polynomial can be a monomial or the sum of monomials. Polynomials are classified by the number of terms they contain. A monomial has one term, a binomial has two terms, and a trinomial has three terms. 8xy - 5x y z, for example, is a binomial. Polynomials are also classified by their degree. The degree of a polynomial is the greatest value among the sums of the exponents on the variables in each term.. EXPLAIN Classifying Polynomials AVOID COMMON ERRORS The binomial 8xy - 5x y z has two terms. The variables in the first term are x and y. The exponent on x is, and the exponent on y is. The number 8 is not a variable, so it has a degree of 0. The first term has a degree of =. The degree of the second term is = 7. Therefore, 8xy - 5x y z is a 7 th degree binomial. Module Students often confuse the degree of a polynomial with the number of terms. Students may be less likely to confuse the two if they equate the word degree with a phrase related to exponents such as maximum power. Lesson PROFESSIONAL DEVELOPMENT A_MNLESE6887_U7M7L.indd 806 Math Background Polynomials are in many ways analogous to counting numbers. Because our number system is base 0, all counting numbers can be written in terms of powers of 0. For example: 65 = = Replacing each of the 0s with a variable creates a polynomial: x + 5 x + x 0 = 6x + 5x +. In counting numbers, the only permissible multipliers of the powers of 0 are the digits 0 through 9. For polynomials, any real number can be a multiplier of the variable terms. This analogy demonstrates how diverse mathematical concepts can share underlying structures. //7 6:0 PM QUESTIONING STRATEGIES Why is the degree of a constant always zero? Think of the constant term attached to a variable with degree 0, which is really. For example, think of the constant 5 as 5x 0 which is really 5, or 5. This shows that the degree of a constant is 0. Understanding Polynomial Expressions 806

3 CONNECT VOCABULARY Prefixes such as mono-, bi-, tri-, and poly- occur often in English. For example, the word monotonous means having one tone and the word bisect means cut into two. In the native languages of some English learners, these prefixes do not occur, so the meanings of monomial, binomial, trinomial, and polynomial may not be familiar. Try using language that may be more familiar, such as saying one term instead of monomial when classifying various polynomials. Example 7 x - 5 x y Classify each polynomial by its degree and the number of terms. Find the degree of each term by adding the exponents of the variables in that term. The greatest degree is the degree of the polynomial. The degree of the term -5 x y is 6, which you obtain by adding the exponents of x and y: 6 = +. Numbers have degree 0. 7 x - 5 x y Degree : 6 7 x has degree, and -5 x y has degree 6 = +. Binomial + n + 8n + n + 8n There are two terms. Degree: has degree 0, n has degree, and 8n has degree. EXPLAIN Writing Polynomials in Standard Form QUESTIONING STRATEGIES Why is the coefficient of the term with the greatest degree also the leading coefficient of the polynomial? When a polynomial is written in standard form, the first term has the greatest degree. Its coefficient is the leading coefficient. INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Discuss with students what pattern to look for to determine whether a polynomial is written in standard form. A polynomial is in standard form when written so that the exponents of a variable decrease from left to right. Thus, the coefficient of the first term is called the leading coefficient because it leads the polynomial. Point out that the leading coefficient is not the largest coefficient and that it can be positive or negative. AVOID COMMON ERRORS Some students may arrange a polynomial using the values of the coefficients instead of the degrees of the terms. Remind them to arrange the terms in order from greatest degree to least degree. Trinomial There are terms. Reflect. What is the degree of 5 x 0 y 0 + 5? 5 x 0 y = = 0, a number with a degree of Is 5 x 0 y a polynomial? Justify your answer. No, the variable y has an exponent of 0.5, which is a decimal number, not a whole number. Your Turn Classify each polynomial by its degree and the number of terms. 5. x y + xy + 5xy 6. 8ab - a b 4 th degree trinomial rd degree binomial Explain Writing Polynomials in Standard Form The terms of a polynomial may be written in any order, but when a polynomial contains only one variable there is a standard form in which it can be written. The standard form of a polynomial containing only one variable is written with the terms in order of decreasing degree. The first term will have the greatest degree, the next term will have the next greatest degree, and so on, until the final term, which will have the lowest degree. When written in this form, the coefficient of the first term is called the leading coefficient. 5 x x + x - is a 4 th degree polynomial written in standard form. It consists of one variable, and its first term is 5 x 4. The leading coefficient is 5 because it is in front of the highest-degree term. Module Lesson COLLABORATIVE LEARNING Peer-to-Peer Activity Have each student work with a partner to write examples of st degree through 6th degree monomials, binomials, and trinomials. Suggest that they create a table to organize their work. Encourage students to write some polynomials with more than one variable per term. Have partners exchange their polynomials so that each one can check that the other s polynomials fit the descriptions. 807 Lesson 7.

4 Example 0x - 4 x + - x Write each polynomial in standard form. Then give the leading coefficient. Find the degree of each term and then arrange them in descending order of their degree. 0x -4 x + - x = -4 x - x + 0x + Degree: 0 0 The standard form is -4 x - x + 0x +. The leading coefficient is -4. z - z 6 + 4z Find the degree of each term and then arrange them in descending order of their degree. Degree: z - z 6 +4z = - z 6 + z + 4z The standard form is - z 6 + z + 4z. The leading coefficient is -. Your Turn 6 6 Write each polynomial in standard form. Then give the leading coefficient x + x x 8. 8 y 5 - y 8 + 0y x x - x + 0, - y y 5 + 0y, - EXPLAIN Simplifying Polynomials CONNECT VOCABULARY English learners may know the word like when referring to a preference, but may be unfamiliar with the phrases like terms and unlike terms. Have students demonstrate their understanding of like and unlike terms by using items that belong to the same category. apples + oranges (unlike) cats + dogs (unlike) apples + apples (like) cats + 4 cats (like) 9. 0x x 0. - b + b b + b x + 0x +, -5b b - b + b, Explain Simplifying Polynomials Polynomials are simplified by combining like terms. Like terms are monomials that have the same variables raised to the same powers. Unlike terms have different powers. Like Terms: Same variable Same power r + 5r + r Unlike Terms: Different power QUESTIONING STRATEGIES What mathematical property allows you to rearrange the terms of a polynomial? The Commutative Property of Addition allows you to switch the order of two addends. What does rearrangement of the terms accomplish? The rearrangement groups like terms with one another so they can be combined. What mathematical property allows you to combine the like terms of a polynomial? The Distributive Property of Addition allows you to factor out a common factor of like terms so that you can simplify the polynomial. Module Lesson DIFFERENTIATE INSTRUCTION Kinesthetic Experience Divide students into groups of four, and have each group member write a monomial on an index card in large print. Students may write a constant or a monomial with a variable. Tell them the variable must be x, but it can be raised to any power between and 4, and it can have any coefficient. Then have the students in each group stand and arrange themselves to form a polynomial written in standard form. If two students use the same power of x, the student who has the larger coefficient should stand first. Have students identify the leading coefficient of the polynomial. Understanding Polynomial Expressions 808

5 INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Make sure that students understand how the Distributive Property is used to simplify polynomials. Give an example using simple numbers to demonstrate, such as 5 (4) + (4) = 4 (5 + ). Students should recognize that the common factor, 4, can be taken out and distributed over the sum of the other two factors. AVOID COMMON ERRORS In determining the degree of a polynomial, some students forget to count for the exponent of a variable that has no visible exponent. Remind them that x = x, so, it has degree. Identify like terms and combine them using the Distributive Property. Simplify. r + 5 r + r ( r + r ) + 5 r Identify like terms by grouping them together in parentheses. r ( + ) + 5 r Combine using the Distributive Property. r + 5 r Simplify. Example Combine like terms to simplify each polynomial. - y - 8 y + y + y - y + y -8 y + y Rearrange in descending order of exponents. (- y + y ) + (-8 y + y ) Group like terms. y (- + ) + y (-8 + ) Combine using the Distributive Property. y (0) + y (-7) Simplify. -7 y p q - 4 p 5 q 4-4 p q + p 5 q 4-4 p 5 q 4 + p 5 q 4 + p q - 4 p q (-4 p 5 q 4 + p 5 q 4 ) + ( p q - 4 p q ) -4 + Rearrange in descending order of exponents. Group like terms. p 5 q 4 ( ) + p q ( ) Combine using the Distributive Property. - - p 5 q 4 ( ) + p q ( ) Simplify. - p 5 q 4 - p q - 4 Reflect. Can you combine like terms without formally showing the Distributive Property? Explain. You do not have to show the Distributive Property to use it. In the initial example, you could have simplified with either of these methods, where you place the polynomial in standard form and add coefficients or just identify like terms and add coefficients. r + 5 r + r 5 r + r + r 5 r + r or r + 5 r + r 5 r + r Both methods depend on the Distributive Property, but its use is not obvious. Module Lesson 809 Lesson 7.

6 Your Turn Simplify.. p q - p q + 4 p q - p q + pq. (a + b) - 6 (b + c) + 8 (a - c) - q p + pq 4. ab - a + 4-5ab + a + 0 ab - a + 4-5ab + a + 0 (- a + a ) + (ab - 5ab) + (6 + 0) Explain 4 Evaluating Polynomials Given a polynomial expression describing a real-world situation and a specific value for the variable(s), evaluate the polynomial by substituting for the variable(s). Then interpret the result. Example 4 p q - p q + 4 p q - p q + pq - q p + 4 q p + p q - p q + pq (- q p + 4 q p ) + (pq) q p (- + 4) + pq a - 4ab + 6 Evaluate the given polynomial to find the solution in each real-world scenario. A skyrocket is launched from a 6-foot-high platform with an initial speed of 00 feet per second. The polynomial -6 t + 00t + 6 gives the height in feet that the skyrocket will rise in t seconds. How high will the rocket rise if it has a 5-second fuse? -6 t + 00t + 6 Write the expression. -6(5) + 00 (5) + 6 Substitute 5 for t. -6(5) + 00 (5) + 6 Simplify using the order of operations (a + b) - 6 (b + c) + 8 (a - c) a + b - 6b - 6c + 8a - 8c a + 8a + b - 6b - 6c - 8c a - b - 4c EXPLAIN 4 Evaluating Polynomials INTEGRATE TECHNOLOGY Students can use a calculator to check that they have evaluated each expression correctly. QUESTIONING STRATEGIES When evaluating an expression, what steps do you follow? Explain why. Use the order of operations. First, perform operations in parentheses or other grouping symbols. Second, simplify powers. Third, perform all multiplication and division from left to right. Fourth, perform all addition and subtraction from left to right. If you don t do the steps in order, you may end up with an answer that is incorrect. The rocket will rise 606 feet. Lisa wants to measure the depth of an empty well. She drops a ball from a height of feet into the well and measures how long it takes the ball to hit the bottom of the well. She uses a stopwatch, starting when she lets go of the ball and ending when she hears the ball hit the bottom of the well. The polynomial -6 t + 0t + gives the height of the ball after t seconds where 0 is the initial speed of the ball and is the initial height the ball was dropped from. Her stopwatch measured a time of. seconds. How deep is the well? (Neglect the speed of sound and air resistance). -6 t + 0t + -6 (.) + -6 (4.84) Write the expression. Substitute. for t. Simplify using order of operations. The ball goes feet below the ground. The well is feet deep. Module 7 80 Lesson LANGUAGE SUPPORT Connect Vocabulary To help students remember the new terminology for this lesson, remind them of other words they know that share these Greek and Latin prefixes: polynomial (poly- means many) can be one monomial, or the sum or difference of more than one monomial monomial (mono- means one) has only one term binomial (bi- means two) has two terms trinomial (tri- means three) has three terms Understanding Polynomial Expressions 80

7 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP. Discuss the fact that the lowest degree a polynomial can have is 0. Be sure to explain that a polynomial can be a monomial, and a monomial can be a number. A polynomial that is a number has no variables or exponents, so its degree is 0. Your Turn Solve each real-world scenario. 5. Nate s client said she wanted the width w of every room in her house increased by feet and the length w decreased by 5 feet. The polynomial (w - 5)(w + ) or w - w - 0 gives the new area of any room in the house. The current width of the kitchen is 6 feet. What is the area of the new kitchen? w - w - 0 (6) - (6) - 0 = 486 The area of the new kitchen is 486 square feet. 6. A skyrocket is launched from a 0-foot-high platform, with an initial speed of 00 feet per second. If the polynomial -6 t + 00t + 0 gives the height that the rocket will rise in t seconds, how high will a rocket with a 4-second fuse rise? -6 t + 00t (4) + 00 (4) + 0 = 564 The rocket will rise 564 feet. SUMMARIZE THE LESSON How do you simplify a polynomial? First, rearrange it in descending order of exponents. Then, identify like terms and group them together. Next, use the Distributive Property to factor out a term that is common to each like term. Add or subtract the numbers that are left over. Finally, write the simplified expression. Image Credits: Peter Barrett/Shutterstock Elaborate 7. What is the degree of the expression -6 t + 00t + 0, where t is a variable? What is the degree of the expression if t =? Are the expressions monomials, binomials, or trinomials? When t is a variable, the expression is a nd degree trinomial. When t =, the polynomial is evaluated at and it is a constant. A constant is a monomial of degree Two cars drive toward each other along a straight road at a constant speed. The distance between the cars is l - ( r + r ) t, where r and r are their speeds, t is a variable representing time and l is the length of their original separation. Write the expression in standard form. What is its degree? What is its leading coefficient? -( r + r ) t + l; Degree ; -( r + r ) is its leading coefficient. 9. The polynomial -6 t + 00t + 0 gives the height of a projectile launched with an initial speed of 00 feet per second t seconds after launch. A second projectile is launched at the same time but with an initial speed of 00 feet per second, with its height given by the polynomial -6 t + 00t + 0. How much higher will the second projectile be than the first after 0 seconds? Evaluating each expression for t = 0 yields a difference of 000, so the second projectile will be 000 feet higher than the first after 0 seconds. 0. Essential Question Check-In What do you have to do to simplify sums of polynomials? What property do you use to accomplish this? You simplify sums of polynomials (which are also polynomials) by combining like terms. You combine like terms using the Distributive Property. Module 7 8 Lesson 8 Lesson 7.

8 Evaluate: Homework and Practice EVALUATE. Is (5 + 4 x 0 )x a monomial? What about (5 + 4 x )x? (5 + 4 x 0 ) x (5 + 4 x ) x (5 + 4) x x (5) + x (4 x ) (9) x 0x + 8 x 8x. Is the sum of two monomials always a monomial? Is their product always a monomial? Their sum is not always a monomial. For example x + y is not a monomial. The product of two monomials will always be a monomial. Online Homework Hints and Help Extra Practice ASSIGNMENT GUIDE (5 + 4 x 0 ) x is a monomial, but (5 + 4 x ) x is a binomial. Classify each polynomial by its degree and the number of terms.. x - 5 x 4. x - x 4 + y x rd degree binomial 5. a 4b - a b + a b x _ 7 th degree trinomial 5 th degree trinomial st degree binomial 7. x + y + z 8. a 5 + b + a b Concepts and Skills Explore Identifying Monomials Example Classifying Polynomials Example Writing Polynomials in Standard Form Practice Exercises, Exercises 8 Exercises 9, st degree trinomial 5 th degree trinomial Write each polynomial in standard form. Then give the leading coefficient. Example Simplifying Polynomials Exercises 6, 4 9. x - 40 x - x 0. + c - c -40 x - x + x; c + c + ; - Example 4 Evaluating Polynomials Exercises 7 0,, 5. b - b + b. 4a - a b - b; 4 a + 7; Simplify each polynomial.. - y - y + y + y 4. - y x - y x + y + y x + y - y - y + y + y - y x - y x + y + y x + y (y - y ) + (- y + y ) (- y x + y x) - y x + ( y + y ) y ( - ) + y (- + ) y x (- + ) - y x + y ( + ) y (-) + y (0) y x (0) - y x + y () - y - y x + y COLLABORATIVE LEARNING Have students work in small groups. Give each group a set of cards that each show a monomial, binomial, trinomial, or polynomial with more than three terms. Include some like terms and do not write them in standard form. Also give the group cards with the words monomial, binomial, trinomial, and polynomial. Have students match each polynomial to the correct word. Then have the students in each group simplify the polynomials by combining like terms and writing them in standard form. Module 7 8 Lesson Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices Recall of Information MP. Reasoning Skills/Concepts MP. Logic 6 Recall of Information MP. Reasoning 7 9 Skills/Concepts MP.4 Modeling 0 Strategic Thinking MP.4 Modeling Skills/Concepts MP. Logic Understanding Polynomial Expressions 8

9 AVOID COMMON ERRORS When determining the degree of a polynomial, some students tally all of the exponents in all of the terms rather than choosing the single term with the greatest sum of exponents. Remind students that the degree of a polynomial is equal to the highest degree of any monomial in the polynomial. 5. xyz + 5 xyz + 0 xy xyz + 5 xyz + 0 xy xyz ( + 5 ) + 0 xy or ( + ) xyz + 04xy Use the information to solve the problem. 6. a + a + ab a + a + ab It is already simplified. 7. Persevere in Problem Solving Lisa is measuring the depth of a well. She drops a ball from a height of h feet into the well and measures how long it takes the ball to hit the bottom of the well. The polynomial -6 t + 0t + h models this situation, where 0 is the initial speed of the ball and h is the height it was dropped from. (This is a different well from the problem you solved before.) She raises her arm very high and drops the ball from a height of 6.0 feet. Her stopwatch measured a time of.5 seconds. How deep is the well? -6 t + 0t + h -6 (.5) (.5) The well is 90 feet deep. 8. Multi-Step Claire and Richard are both artists who use square canvases. Claire uses the polynomial 50 x + 50 to decide how much to charge for her paintings, and Richard uses the polynomial 40 x + 50 to decide how much to charge for his paintings. In each polynomial, x is the height of the painting in feet. a. How much does Claire charge for a 6-foot painting? 50 (6) + 50 = = $050 b. How much does Richard charge for a 5-foot painting? 40 (5) + 50 = = $50 c. To the nearest tenth, for what height will both Claire and Richard charge the same amount for a painting? Explain how to find the answer. You set the expressions equal and solve. 50 x + 50 = 40 x x = 00 x = 0 x = 0. feet d. When both Claire and Richard charge the same amount for a painting, how much does each charge? 40 (0) + 50 = = $750 Module 7 8 Lesson Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices Strategic Thinking MP. Logic Skills/Concepts MP. Logic 4 Strategic Thinking MP. Logic 5 Skills/Concepts MP. Reasoning 8 Lesson 7.

10 9. Make a Prediction The number of cells in a bacteria colony increases according to a polynomial expression that depends on the temperature. The expression for the number of bacteria is t + 4t + 4 when the temperature of the colony is 0 C and t + t + 4 when the colony grows at 0 C. t represents the time in seconds that the colony grows at the given temperature. a. After minute, will the population be greater in a colony at 0 C or 0 C? Explain. t + 4t + 4 t + t + 4 (60) + 4 (60) + 4 (60) + (60) When t is 60 seconds, the colony at 0 C will have 60 more bacteria. b. After 0 minutes, how will the colonies compare in size? Explain. The colony at 0 C will have 600 more bacteria after 0 minutes. GRAPHIC ORGANIZERS Divide students into groups of three or four and have them create posters to reinforce their understanding of polynomials. Have half the groups make posters titled Polynomials and the other half make posters titled Not Polynomials. Each poster should include the definition of a polynomial and give three examples of polynomials or non-polynomials. Have the group that writes polynomials write each one in standard form and identify its degree. Have the group that writes non-polynomials explain why each example is not a polynomial. c. After 000 minutes, how will the colonies compare in size? Will one colony always have more bacteria? Explain. The colony at 0 C will have 60,000 more bacteria after 000 minutes. After t = 0, the 0 C colony will always have t more bacteria. 0. Two cars are driving toward each other along a straight road. Their separation distance is l - ( r + r ) t, where l is their original separation distance and r and r are their speeds. Will the cars meet? When? What if they are going in the same direction and not driving toward one another? Will they meet then? l l - ( r + r ) t = 0 l = ( r + r ) t r + r = t If they are going in the same direction, you can change the sign of one of l the rates t = r - r r - r > 0 r > r. If both cars are going to the l right, the one on the left has to go faster and they will meet at t = r - r. If they are going the same speed, however, t will be undefined or negative and they will never meet. Image Credits: Photo Researchers/Getty Images Module 7 84 Lesson Understanding Polynomial Expressions 84

11 VISUAL CUES For many students it is easier to identify the degree of each term in a polynomial when every term is written with the variables and their exponents. Encourage students to look at each term of a polynomial. If a variable does not have an exponent, have them write a as the exponent. Then have them circle each exponent and add the exponents of the variables in a term to find the degree of the term. Repeat the procedure for each term in a polynomial. The greatest degree among the terms is the degree of the polynomial. JOURNAL Have students explain how to classify a polynomial according to its degree and number of terms. H.O.T. Focus on Higher Order Thinking. Explain the Error Enrique thinks that the polynomial x + x + 4 has a degree of 4 since + = + = 4. Explain his error and determine the correct degree. Enrique treated the numbers like variables, but their degree is 0. The degree of the polynomial is from the exponent of over the x in the term x.. Analyze Relationships Sewell is doing a problem regarding the area of pairs of squares. Sewell says that the expression (x + ) will be greater than (x - ) for all values of x because x + will always be greater than x -. Why is he correct when the expressions are areas of squares? Is he correct for any real x outside this model? If x = 0, they are equal. (0 + ) = () = (0 - ) = (-) = If x < 0, then (x + ) < (x - ). The expressions are areas of squares, which are strictly positive. The lengths of the sides are x + and x -, which are also strictly positive. Thus, x - > 0 x >. For these values of x, the expression (x + ) > (x - ) will always be true. He is correct in this model but not correct for all real values of x.. Counterexamples Prove by counterexample that the sum of monomials is not necessarily a monomial. By counterexample, x + y is not a monomial, although it is a sum of monomials. 4. Communicate Mathematical Ideas Polynomials are simplified by combining like terms. When combining like terms, you use the Distributive Property. Prove that the Distributive Property, a (b + c) = a b + a c, holds over the positive integers a, b, c > 0 from the definition of multiplication: a b = a + a + + a. 5. Analyze Relationships A right triangle has height h and base h + 8. Write an expression that represents the area of the triangle. Then calculate the area of a triangle with a height of 6 cm. h (h + 8) ; area = 9 c m b times a b + a c = a + a + + a + a + a + + a = a + a + + a = a (b + c) b times c times b + c times The a variables can be shuffled around using the Associative and Commutative Properties of the integers. Induction should be used to be very rigorous. Module 7 85 Lesson 85 Lesson 7.

12 Lesson Performance Task A pyrotechnics specialist is designing a firework spectacular for a company s 75 th anniversary celebration. She can vary the launch speed to 00, 50, 00, or 400 feet per second, and can set the fuse on each firework for, 4, 5, or 6 seconds. Create a table of the various heights the fireworks can explode at if the height of the firework is modeled by the function h (t) = -6 t + v 0 t, where t is the time in seconds and v 0 is the initial speed of the firework. Design a fireworks show using firing heights and at least 0 fireworks. Have some fireworks go off simultaneously at different heights. Describe your display so you will know what needs to be launched and when they will go off. The polynomials are as follows: h A (t) = -6 t + 00t h B (t) = -6 t + 50t CONNECT VOCABULARY Some students may not be familiar with the terms in this Lesson Performance Task. Have a student volunteer describe pyrotechnics, launch speed, and fuse. Pyrotechnics is the art of making fireworks. Launch speed is the speed of each firework at the moment it is fired. A fuse is a flammable cord that is lit in order to set off an explosive. INTEGRATE TECHNOLOGY Students can use a calculator to check that they have evaluated the polynomials correctly. h C (t) = -6 t + 00t h D (t) = -6 t + 400t The table shows the value of each polynomial evaluated at, 4, 5, and 6 seconds. The method of evaluation is shown below. h A () = -6() + 00 () = -6(9) = = 456 Launch Speed Second Fuse 4 Second Fuse 5 Second Fuse 6 Second Fuse The remainder of the answers will vary, but all answers should involve a list of when the fireworks will go off, the speeds fired, and the fuse settings. Answers should also include a second list showing which fireworks are Image Credits: manzrussali/shutterstock exploding every second and their heights. Module 7 86 Lesson EXTENSION ACTIVITY Ask students to consider the path of a firework that goes up and comes back down without exploding. Have them use graphing calculators, trial and error, or equation solving to find the maximum height that a firework reaches with each launch speed. Students can analyze each firework s height after t seconds by entering the functions on a graphing calculator. They can use the TRACE or TABLE features to find the maximum height. The fireworks launched at 00, 50, 00, and 400 feet per second reach maximum heights of 65, 976, 406, and 500 feet, respectively. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Understanding Polynomial Expressions 86