# Maintaining Mathematical Proficiency

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1 Chapter 7 Maintaining Mathematical Proficiency Simplify the expression. 1. 5x 6 + 3x. 3t + 7 3t s 4 + 4s 6 5s 4. 9m m 3 + 5m p 7 3p 4 1 z ( ) 7. 6( x + ) ( h + 4) 3( h 4) 9. 7( z + 4) 3( z + ) ( z 3) Find the greatest common factor , , , , , , Explain how to find the greatest common factor of 4, 70, and

2 Name Date 7.1 Adding and Subtracting Polynomials For use with Exploration 7.1 Essential Question How can you add and subtract polynomials? 1 EXPLORATION: Adding Polynomials Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Write the expression modeled by the algebra tiles in each step. Step 1 + ( 3x + ) + ( x 5) Step Step 3 Step 4 EXPLORATION: Subtracting Polynomials Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Write the expression modeled by the algebra tiles in each step. Step 1 ( x + x + ) ( x 1 ) Step + Step 3 04 Algebra 1 Copyright Big Ideas Learning, LLC

3 7.1 Adding and Subtracting Polynomials (continued) EXPLORATION: Subtracting Polynomials (continued) Step 4 Step 5 Communicate Your Answer 3. How can you add and subtract polynomials? 4. Use your methods in Question 3 to find each sum or difference b. ( 4x + 3) + ( x ) a. ( x x 1) ( x x 1) c. ( x + ) ( 3x + x + 5) d. ( x 3x) ( x x + 4) 05

4 Name Date 7.1 Notetaking with Vocabulary For use after Lesson 7.1 In your own words, write the meaning of each vocabulary term. monomial degree of a monomial polynomial binomial trinomial degree of a polynomial standard form leading coefficient closed Notes: 06 Algebra 1 Copyright Big Ideas Learning, LLC

5 7.1 Notetaking with Vocabulary (continued) Core Concepts Polynomials A polynomial is a monomial or a sum of monomials. Each monomial is called a term of the polynomial. A polynomial with two terms is a binomial. A polynomial with three terms is a trinomial. Binomial Trinomial 5x + x + 5x + The degree of a polynomial is the greatest degree of its terms. A polynomial in one variable is in standard form when the exponents of the terms decrease from left to right. When you write a polynomial in standard form, the coefficient of the first term is the leading coefficient. leading coefficient degree x 3 + x 5x + 1 constant term Notes: Extra Practice In Exercises 1 8, find the degree of the monomial. 1. 6s. w abc x y 6. 4r st 7. 10mn

6 Name Date 7.1 Notetaking with Vocabulary (continued) In Exercises 9 1, write the polynomial in standard form. Identify the degree and leading coefficient of the polynomial. Then classify the polynomial by the number of terms x + 3x y 11. 3x + 6x 1. f f + f In Exercises 13 16, find the sum. 13. ( 4x + 9) + ( 6x 14) 14. ( 3a ) + ( 7a + 5) 15. ( x + 3x + 5) + ( x + 6x 4) 16. ( t + 3t 3 3) + ( t + 7t t 3 ) In Exercises 17 0, find the difference. 17. ( g 4) ( 3g 6) 18. ( 5h ) ( 7h + 6) 19. ( x 5) ( 3x x 8) 0. ( k + 6k 3 4) ( 5k 3 + 7k 3k ) 08 Algebra 1 Copyright Big Ideas Learning, LLC

7 7. Multiplying Polynomials For use with Exploration 7. Essential Question How can you multiply two polynomials? 1 EXPLORATION: Multiplying Monomials Using Algebra Tiles Work with a partner. Write each product. Explain your reasoning. = = a. b. = = c. d. = = e. f. = = g. h. = = i. j. 09

8 Name Date 7. Multiplying Polynomials (continued) EXPLORATION: Multiplying Binomials Using Algebra Tiles Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Write the product of two binomials modeled by each rectangular array of algebra tiles. In parts (c) and (d), first draw the rectangular array of algebra tiles that models each product. a. ( x + 3)( x ) = b. ( x )( x ) = c. ( x + )( x 1 ) = d. ( x )( x ) 3 = Communicate Your Answer 3. How can you multiply two polynomials? 4. Give another example of multiplying two binomials using algebra tiles that is similar to those in Exploration. 10 Algebra 1 Copyright Big Ideas Learning, LLC

9 7. Notetaking with Vocabulary For use after Lesson 7. In your own words, write the meaning of each vocabulary term. FOIL Method Core Concepts FOIL Method To multiply two binomials using the FOIL Method, find the sum of the products of the First terms, (x + 1)(x + ) x(x) = x Outer terms, (x + 1)(x + ) x() = x Inner terms, and (x + 1)(x + ) 1(x) = x Last terms. (x + 1)(x + ) 1() = ( )( ) x + 1 x + = x + x + x + = x + 3x + Notes: 11

10 Name Date 7. Notetaking with Vocabulary (continued) Extra Practice In Exercises 1 6, use the Distributive Property to find the product. 1. ( x )( x 1). ( b 3)( b + ) 3. ( g + )( g + 4) 4. ( a 1)( a + 5) 5. ( 3n 4)( n + 1) 6. ( r + 3)( 3r + ) In Exercises 7 1, use a table to find the product. 7. ( x 3)( x ) 8. ( y + 1)( y 6) 9. ( q + 3)( q + 7) 10. ( w 5)( w 3) 11. ( 6h )( 3 h) 1. ( 3 + 4j)( 3j + 4) 1 Algebra 1 Copyright Big Ideas Learning, LLC

11 7. Notetaking with Vocabulary (continued) In Exercises 13 18, use the FOIL Method to find the product. 13. ( x + )( x 3) 14. ( z + 3)( z + ) 15. ( h )( h + 4) 16. ( m 1)( m + ) 17. ( 4n 1)( 3n + 4) 18. ( q 1)( q + 1) In Exercises 19 4, find the product. 19. ( x )( x + x 1) 0. ( a)( 3a + 3a 5) 1. ( h + 1)( h h 1). ( d 3)( d 4d 1) ( 3n + n 5)( n + 1) 4. ( p + p 3)( 3p 1) 13

12 Name Date 7.3 Special Products of Polynomials For use with Exploration 7.3 Essential Question What are the patterns in the special products a + b a b, a + b, and a b? ( )( ) ( ) ( ) 1 EXPLORATION: Finding a Sum and Difference Pattern Work with a partner. Write the product of two binomials modeled by each rectangular array of algebra tiles. a. ( x + )( x ) = b. ( x )( x ) = EXPLORATION: Finding the Square of a Binomial Pattern Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Draw the rectangular array of algebra tiles that models each product of two binomials. Write the product. a. ( x + ) = b. ( ) x 1 = 14 Algebra 1 Copyright Big Ideas Learning, LLC

13 7.3 Special Products of Polynomials (continued) Communicate Your Answer a + b a b, a + b, 3. What are the patterns in the special products ( )( ) ( ) and a ( b)? 4. Use the appropriate special product pattern to find each product. Check your answers using algebra tiles. a. ( x + 3)( x 3) b. ( x 4)( x + 4) c. ( 3x + 1)( 3x 1) d. ( x + 3) e. ( x ) f. ( 3x + 1) 15

14 Name Date 7.3 Notetaking with Vocabulary For use after Lesson 7.3 In your own words, write the meaning of each vocabulary term. binomial Core Concepts Square of a Binomial Pattern Algebra Example ( ) a b a ab b + = + + ( x + 5) = ( x) + ( x)( 5) + ( 5) = x + x ( ) a b = a ab + b ( x 3) = ( x) ( x)( 3) + ( 3) Notes: = + 4x 1x 9 Sum and Difference Pattern Algebra Example ( a + b)( a b) = a b ( x )( x ) x Notes: = 9 16 Algebra 1 Copyright Big Ideas Learning, LLC

15 7.3 Notetaking with Vocabulary (continued) Extra Practice In Exercises 1 18, find the product. 1. ( a + 3). ( b ) 3. ( c + 4) 4. ( x + 1) 5. ( 3x ) 6. ( 4p 3) 7. ( 3x + y ) 8. ( a 3b ) 9. ( 4c + 5d) 10. ( x 3)( x + 3) 11. ( q + 5)( q 5) 1. ( t 11)( t + 11) 17

16 Name Date 7.3 Notetaking with Vocabulary (continued) 13. ( 5 1)( 5 + 1) a a b + 1 b c + c ( m + n)( m n) 17. ( 3j k)( 3j k) a + b 6a + b In Exercises 19 4, use special product patterns to find the product ( 31 ) 3. ( 0.7 ) 4. ( 109 ) 5. Find k so that kx 1x + 9 is the square of a binomial. 18 Algebra 1 Copyright Big Ideas Learning, LLC

17 7.4 Solving Polynomial Equations in Factored Form For use with Exploration 7.4 Essential Question How can you solve a polynomial equation? 1 EXPLORATION: Matching Equivalent Forms of an Equation Work with a partner. An equation is considered to be in factored form when the product of the factors is equal to 0. Match each factored form of the equation with its equivalent standard form and nonstandard form. Factored Form Standard Form Nonstandard Form x 1 x 3 = 0 A. x x = 0 1. x 5x = 6 a. ( )( ) x x 3 = 0 B. + = 0 b. ( )( ) c. ( )( ) x x. ( x ) 1 = 4 x + 1 x = 0 C. x 4x + 3 = 0 3. x x = x 1 x + = 0 D = 0 d. ( )( ) x x 4. xx ( ) + 1 = x + 1 x 3 = 0 E. x x 3 = 0 5. e. ( )( ) x 4x = 3 EXPLORATION: Writing a Conjecture Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Substitute 1,, 3, 4, 5, and 6 for x in each equation and determine whether the equation is true. Organize your results in the table. Write a conjecture describing what you discovered. a. ( x )( x ) Equation x =1 x = x = 3 x = 4 x = 5 x = 6 1 = 0 b. ( x )( x ) 3 = 0 c. ( x )( x ) 3 4 = 0 d. ( x )( x ) 4 5 = 0 e. ( x )( x ) 5 6 = 0 f. ( x )( x ) 6 1 = 0 19

18 Name Date 7.4 Solving Polynomial Equations in Factored Form (continued) 3 EXPLORATION: Special Properties of 0 and 1 Work with a partner. The numbers 0 and 1 have special properties that are shared by no other numbers. For each of the following, decide whether the property is true for 0, 1, both, or neither. Explain your reasoning. a. When you add to a number n, you get n. b. If the product of two numbers is, then at least one of the numbers is 0. c. The square of is equal to itself. d. When you multiply a number n by, you get n. e. When you multiply a number n by, you get 0. f. The opposite of is equal to itself. Communicate Your Answer 4. How can you solve a polynomial equation? 5. One of the properties in Exploration 3 is called the Zero-Product Property. It is one of the most important properties in all of algebra. Which property is it? Why do you think it is called the Zero-Product Property? Explain how it is used in algebra and why it so important. 0 Algebra 1 Copyright Big Ideas Learning, LLC

19 7.4 Notetaking with Vocabulary For use after Lesson 7.4 In your own words, write the meaning of each vocabulary term. factored form Zero-Product Property roots repeated roots Core Concepts Zero-Product Property Words If the product of two real numbers is 0, then at least one of the numbers is 0. Algebra If a and b are real numbers and ab = 0, then a = 0 or b = 0. Notes: 1

20 Name Date 7.4 Notetaking with Vocabulary (continued) Extra Practice In Exercises 1 1, solve the equation. 1. xx+ ( 5) = 0. aa ( 1) = 0 3. ( ) 5pp = 0 4. ( c )( c + 1) = 0 5. ( b 6)( 3b + 18) = 0 6. ( s)( s) = 0 7. ( x 3) = 0 8. ( 3d + 7)( 5d 6) = 0 9. ( t )( t ) = ( w + 4) ( w + 1) = g( g)( g) + = 1. ( m )( m)( m) = 0 3 Algebra 1 Copyright Big Ideas Learning, LLC

21 7.4 Notetaking with Vocabulary (continued) In Exercises 13 18, factor the polynomial x + 3x 14. 4y 0y u 6u z + z 17. 4h + 8h f 45 f In Exercises 19 4, solve the equation k + k = n 49n = z + 5z = 0. 6x 7 = x 3. s = 11s 4. 7p = 1p 5. A boy kicks a ball in the air. The height y (in feet) above the ground of the ball is modeled by the equation y = 16x + 80 x, where x is the time (in seconds) since the ball was kicked. Find the roots of the equation when y = 0. Explain what the roots mean in this situation. 3

22 Name Date 7.5 Factoring x + bx + c For use with Exploration 7.5 Essential Question How can you use algebra tiles to factor the trinomial x + bx + c into the product of two binomials? 1 EXPLORATION: Finding Binomial Factors Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use algebra tiles to write each polynomial as the product of two binomials. Check your answer by multiplying. Sample x + 5x + 6 Step 1 Arrange algebra tiles that Step model x + 5x + 6 into a rectangular array. Use additional algebra tiles to model the dimensions of the rectangle. Step 3 Write the polynomial in factored form using the dimensions of the rectangle. width length Area = x + 5x + 6 = (x + )(x + 3) a. x 3x + = b. x + 5x + 4 = 4 Algebra 1 Copyright Big Ideas Learning, LLC

23 7.5 Factoring x + bx + c (continued) 1 EXPLORATION: Finding Binomial Factors (continued) c. x 7x + 1 = d. x + 7x + 1 = Communicate Your Answer. How can you use algebra tiles to factor the trinomial x + bx + c into the product of two binomials? 3. Describe a strategy for factoring the trinomial x + bx + c that does not use algebra tiles. 5

24 Name Date 7.5 Notetaking with Vocabulary For use after Lesson 7.5 In your own words, write the meaning of each vocabulary term. polynomial FOIL Method Zero-Product Property Core Concepts Factoring x + bx + c When c Is Positive x + bx + c = x + p x + q when p + q = b and pq = c. When c is positive, p and q have the same sign as b. Algebra ( )( ) Examples x + 6x + 5 = ( x + 1)( x + 5) x 6x + 5 = ( x 1)( x 5) Notes: Factoring x + bx + c When c Is Negative x + bx + c = x + p x + q when p + q = b and pq = c. When c is negative, p and q have different signs. Algebra ( )( ) Example x 4 x 5 = ( x + 1 )( x 5 ) Notes: 6 Algebra 1 Copyright Big Ideas Learning, LLC

25 7.5 Notetaking with Vocabulary (continued) Extra Practice In Exercises 1 1, factor the polynomial. 1. c + 8c + 7. a + 16a x + 11x d + 6d s + 11s u + 10u b + 3b y y 9. u + 3u z z h + h 4 1. f 3f 40 7

26 Name Date 7.5 Notetaking with Vocabulary (continued) In Exercises 13 18, solve the equation. 13. g 13g + 40 = k 5k + 6 = w 7w + 10 = x x = r 3r = 18. t 7t = The area of a right triangle is 16 square miles. One leg of the triangle is 4 miles longer than the other leg. Find the length of each leg. 0. You have two circular flower beds, as shown. The sum of the areas of the two flower beds is 136 π square feet. Find the radius of each bed. 8 Algebra 1 Copyright Big Ideas Learning, LLC

27 7.6 Factoring ax + bx + c For use with Exploration 7.6 Essential Question How can you use algebra tiles to factor the trinomial + + ax bx c into the product of two binomials? 1 EXPLORATION: Finding Binomial Factors Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use algebra tiles to write each polynomial as the product of two binomials. Check your answer by multiplying. Sample x + 5x + Step 1 Arrange algebra tiles that Step model x + 5x + into a rectangular array. Use additional algebra tiles to model the dimensions of the rectangle. Step 3 Write the polynomial in factored form using the dimensions of the rectangle. width Area = x + 5x + = (x + )(x + 1) length a. 3x + 5x + = 9

28 Name Date 7.6 Factoring ax + bx + c (continued) 1 EXPLORATION: Finding Binomial Factors (continued) b. 4x + 4x 3 = c. x 11x + 5 = Communicate Your Answer. How can you use algebra tiles to factor the trinomial ax + bx + c into the product of two binomials? 3. Is it possible to factor the trinomial x + x + 1? Explain your reasoning. 30 Algebra 1 Copyright Big Ideas Learning, LLC

29 7.6 Notetaking with Vocabulary For use after Lesson 7.6 In your own words, write the meaning of each vocabulary term. polynomial greatest common factor (GCF) Zero-Product Property Notes: 31

30 Name Date 7.6 Notetaking with Vocabulary (continued) Extra Practice In Exercises 1 18, factor the polynomial. 1. c 14c 36. 4a + 8a x 6x 4 4. d d s + 55s q + 30q g 37g k 11k w + 9w a + 5a b + 14b t + 1t 9 3 Algebra 1 Copyright Big Ideas Learning, LLC

31 7.6 Notetaking with Vocabulary (continued) 13. 1b + 5b x + x g 11g d d r 4r x 14x The length of a rectangular shaped park is ( 3x + 5) miles. The width is ( x + 8) The area of the park is 360 square miles. What are the dimensions of the park? miles. 0. The sum of two numbers is 8. The sum of the squares of the two numbers is 34. What are the two numbers? 33

32 Name Date 7.7 Factoring Special Products For use with Exploration 7.7 Essential Question How can you recognize and factor special products? 1 EXPLORATION: Factoring Special Products Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use algebra tiles to write each polynomial as the product of two binomials. Check your answer by multiplying. State whether the product is a special product that you studied in Section 7.3. a. 4x 1 = b. 4x 4x + 1 = c. 4x 4x = d. 4x 6x + = 34 Algebra 1 Copyright Big Ideas Learning, LLC

33 7.7 Factoring Special Products (continued) EXPLORATION: Factoring Special Products Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use algebra tiles to complete the rectangular arrays in three different ways, so that each way represents a different special product. Write each special product in standard form and in factored form. Communicate Your Answer 3. How can you recognize and factor special products? Describe a strategy for recognizing which polynomials can be factored as special products. 4. Use the strategy you described in Question 3 to factor each polynomial. a. 5x + 10x + 1 b. 5x 10x + 1 c. 5x 1 35

34 Name Date 7.7 Notetaking with Vocabulary For use after Lesson 7.7 In your own words, write the meaning of each vocabulary term. polynomial trinomial Core Concepts Difference of Two Squares Pattern Algebra Example = ( + )( ) x 9 = x 3 = ( x + 3)( x 3) a b a b a b Notes: Perfect Square Trinomial Pattern Algebra Example + + = ( + ) x x x ( x)( ) a ab b a b = ( x 3) = + + = ( ) x x x ( x)( ) a ab b a b Notes: = ( x 3) = 36 Algebra 1 Copyright Big Ideas Learning, LLC

35 7.7 Notetaking with Vocabulary (continued) Extra Practice In Exercises 1 6, factor the polynomial. 1. s 49. t x 4. 4g h k In Exercises 7 1, use a special product pattern to evaluate the expression

36 Name Date 7.7 Notetaking with Vocabulary (continued) In Exercises 13 18, factor the polynomial. 13. x + 16x p + 8p r 6r a 18a c + 84c x 0x + 1 In Exercises 19 4, solve the equation. 19. x 144 = y = c + 14c + 49 = 0. d 4d + 4 = 0 3. n + n = k + 9 = k The dimensions of a rectangular prism are ( x + 1) feet by ( ) prism is ( 4x 1) cubic feet. What is the value of x? x + feet by 4 feet. The volume of the 38 Algebra 1 Copyright Big Ideas Learning, LLC

37 7.8 Factoring Polynomials Completely For use with Exploration 7.8 Essential Question How can you factor a polynomial completely? 1 EXPLORATION: Writing a Product of Linear Factors Work with a partner. Write the product represented by the algebra tiles. Then multiply to write the polynomial in standard form. a. ( )( )( ) b. ( )( )( ) c. ( )( )( ) d. ( )( )( ) e. ( )( )( ) f. ( )( )( ) EXPLORATION: Matching Standard and Factored Forms Work with a partner. Match the standard form of the polynomial with the equivalent factored form on the next page. Explain your strategy. x a. 3 d. 3 + x b. x 4x + 4x e. 3 x x c. 3 x x 3x f. 3 x + x x 3 x x + x g. x x h. 3 4 x + x i. 3 x x 3 j. 3 x 3x x + k. 3 x x 3x + l. 3 x 4x + 3x m. x x n. 3 3 x 4x 4x + + o. 3 x + x + x 39

38 Name Date 7.8 Factoring Polynomials Completely (continued) EXPLORATION: Matching Standard and Factored Forms (continued) A. xx ( + 1)( x 1) B. xx ( 1) C. xx+ ( 1) D. xx ( + )( x 1) E. xx ( 1)( x ) F. xx ( + )( x ) G. xx ( ) H. xx+ ( ) I. x ( x 1) J. x ( x + 1) K. x ( x ) L. x ( x + ) M. xx ( + 3)( x 1) N. xx ( + 1)( x 3) O. xx ( 1)( x 3) Communicate Your Answer 3. How can you factor a polynomial completely? 4. Use your answer to Question 3 to factor each polynomial completely. 3 3 x + 4x + 3x b. x 6x + 9x c. x + 6x + 9x a Algebra 1 Copyright Big Ideas Learning, LLC

39 7.8 Notetaking with Vocabulary For use after Lesson 7.8 In your own words, write the meaning of each vocabulary term. factoring by grouping factored completely Core Concepts Factoring by Grouping To factor a polynomial with four terms, group the terms into pairs. Factor the GCF out of each pair of terms. Look for and factor out the common binomial factor. This process is called factoring by grouping. Notes: Guidelines for Factoring Polynomials Completely To factor a polynomial completely, you should try each of these steps. 1. Factor out the greatest common monomial factor. 3x + 6x = 3x( x + ). Look for a difference of two squares or a perfect square trinomial. x + 4 x + 4 = ( x + ) 3. Factor a trinomial of the form ax + bx + c into a product 3x 5x = 3x + 1 x of binomial factors. ( )( ) 4. Factor a polynomial with four terms by grouping. x 3 + x 4x 4 = ( x + 1)( x 4) Notes: 41

40 Name Date 7.8 Notetaking with Vocabulary (continued) Extra Practice In Exercises 1 8, factor the polynomial by grouping. b 4b + b 4. ac + ad + bc + bd d + c + cd + d 4. 5t + 6t + 5t s + s 64s a + a 30a x 1x 5x h + 18h 35h Algebra 1 Copyright Big Ideas Learning, LLC

41 7.8 Notetaking with Vocabulary (continued) In Exercises 9 16, factor the polynomial completely c 4c x 5x 11. a + 3a 1. 9x + 3x p p x 0x s + s 1s t + t 10t 5 In Exercises 17, solve the equation x 1x + 30 = y 5y = 19. c 81c = d + 9 = d + d 1. 48n 3n = 0. x + 3x = 16x

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