UNIT 9 (Chapter 7 BI) Polynomials and Factoring Name: The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay attention and make any necessary changes. All assignments are due the following class period unless indicated otherwise. Monday Tuesday Wednesday Thursday Friday March 16 Review 17 Unit 8 Test 18 Pretest Section 7.1 19 Section 7. 0 Section 7. 4 6 7 Quiz 7.1-7. Section 7.4 Section 7. a Section 7.b Section 7.c Review End of rd MP 0 1 April 1 Break Begins Section 7.6a Bo Method Section 7.6b Section 7.6c Section 7.7 1 14 1 16 17 Section 7.8 Review Review Unit 9 Test Section Page Assignment 7.1 p. 6 #6- even -7 7. p. 69 #4-44 even, 47,48 7. p. 7 #-4 even, 7-4 Review --- Worksheet in packet 7.4 & a p. 81 #4-40 even, 49-7.b p. 89 #1-, 8-40 even, -8 7.c --- Worksheet in packet 7.6a 7.6b --- Worksheet in the packet 7.6c --- Worksheet in packet 7.7 p. 401 # 7.8 p. 407 # 1
Lesson 7.1 Adding and Subtracting Polynomials Algebra 1 Essential Question How can you add and subtract polynomials? Warm-Up Eercise (a) 4 (b) ( 4) 1 (c) 9 ( 8) 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 The degree of a polynomial is the greatest degree of its terms. A polynomial in one variable is in standard form when the eponents 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. 1. Eamining Polynomials Decide whether or not it is a polynomial. If it is, then write it in standard form, list its degree, leading coefficient, and constant term. (a) (b) (c) 8 (d) Classifying Polynomials Polynomials can be classified according to their degree and by the number of terms. Fill out the chart below. Polynomial Degree Classified by Degree Classified by # of terms A. B. C. D. E. F.
. Classifying Polynomials Classify each polynomial by its degree and by its number of terms. (a) 4 (b) 7 (c) 7. Adding Polynomials Add the polynomials using a vertical format in part (a) and a horizontal format in part (b). (a) 4. Subtracting Polynomials Subtract the polynomials using a vertical format in part (a) and a horizontal format in part (b). (a) (b) (b)
Lesson 7. Multiplying Polynomials Algebra 1 Essential Question How can you multiply two polynomials? Warm-Up Eercise Simplify the Epression. (a) ( ) ( 6) (b) ( 6) (c) ( 1 )7 10 (d) ( ) 1. Investigating Binomial Multiplication Use the diagram at the right and determine the area of the entire region. Then complete the statement below. 1 1 1 =. Multiplying Binomials Using the Distributive Property Find the product using the distributive property. (a) ( )( ) (b) ( )( ) (c) ( )( 1) (d) (4 1)( ) 4
. Multiplying Binomials Using F.O.I.L. Pattern Find the product of binomials using the F.O.I.L. method. (a) ( )( 1) (b) ( 4)( 1) (c) ( )( 4) In Eercises 4 9, use a table to find the product. 4.. y 1 y 6 6. q q 7 7. w w 8. 6h h 9. 4 j j 4
Lesson 7. Special Product of Polynomials Algebra 1 Essential Question What are the patterns in the special products ( a + b) and (a + b)( a b)? Warm-Up Eercise Simplify the Epression. (a) 1 ( ) (b) ( m 6 ) (c) ( y ) (d) ( ) 1. Using Sum and Difference Binomial Patterns Find the product. (a) ( b )(b ) (b) ( d )(d ). Special Product: Squaring Binomials Find the product. (a) ( n 4) (b) ( 7y) (c) 1 ( a 4) 6
7.1-7. Review Worksheet Name Identify the leading coefficient, constant term and classify the polynomial by degree and by number of terms. 1.. 8 + 4. 6 + 4 - Add or subtract. 4. ( - ) (7 + 1). ( + 4) (- + ) 6. ( m + m - 4m ) + (m + m - 4) 7. ( + 1) + (- 7) - ( + ) 8. ( + 4) - ( + 7) + ( -1) 9. ( - 4 + ) ( + 6 1) Find the product. 10. - ( ) 11. ( 4 )( 8) 7
1. ( - )(4 + 1) 1. ( 1)( + 7) 14. 4( )( + 4) 1. ( ½ ) ( 4 - ) 16. ( + 1)( 4-7) 17. ( + 1) ( + ) Write the square of the binomial as a trinomial. 18. ( + ) 19. ( y 4) Find the product. 0. ( + 7)( 7) 1. (m ) (m + ). ( ½ + 4)( ½ 4) 8
Lesson 7.4 The Zero Product Property Algebra 1 Essential Question How can you solve a polynomial equation? 1 EXPLORATION: Writing a Conjecture Work with a partner. Substitute 1,,, 4,, and 6 for in each equation and determine whether the equation is true. Organize your results in the table. Write a conjecture describing what you discovered. a. Equation =1 = = =4 = =6 1 0 b. 0 c. 4 0 d. 4 0 e. 6 0 f. 6 1 0 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. 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 by n by, you get 0. f. The opposite of is equal to itself. One of the properties in Eploration 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? Eplain how it is used in algebra and why it so important. 9
1. Using the Zero Product Property Use the Zero Product Property to solve the equations written in factored form. Factored Form (a) ( 4)( 9) 0 Zero Product Property (b) ( 1)( ) 0 1 (c) )( 1) 0 (d) ( 8) 0 ( 4. Zero Product Property with Special Products Use the Zero Product Property to solve the equations written in factored form. (a) ( 7) 0 (b) ( 1 6) 0 4 (c) ( )( )( 10) 0 (d) ) ( 11)( 1) 0 4 ( 8 10
Lesson 7.a Factoring the Greatest Common Factor (GCF) Algebra 1 Warm-up Eercise Use the Distributive Property. (a) (4 6 9) (b) (7 8 4) (c) 7y( y) Common Monomial Factoring You can think of common monomial factoring as reversing the distributive property. Our goal here is to factor, or pull out, the greatest common monomial factor. We call this factoring the GCF. E. 1. Factor the GCF out of each polynomial. (a) (b). Factor the GCF out of each polynomial. (a) (b) (c) (d) In Eercises -, solve the equation.. 6k k 0 4. n 49n 0. 4z z 0 6. 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 16 80, where is the time (in seconds) since the ball was kicked. Find the roots of the equation when y = 0. Eplain what the roots mean in this situation. 11
7.A Practice: Factoring the Greatest Common Factor Name: In each of the polynomials, factor out the GCF. If there is no GCF, then just write No GCF. 1. 10. 9 4 18. 8 1 4 4. 6v 18v. 4q 4 1q 6. 9 7. 4 t t 6 8. 4a a 6 8a 9. 18d 6d d 10. 10 +4-16 11. 6 8 +90 7 +48 1. 18 8 +10 6-10 1. 6 10-8 -0 14. 40-0 +1 1. 0 8-4 -0 16. 16 8 +14 6-14 7 17. 0 6-40 4-1 8 18. 1 7 +1-90 1
Lesson 7.b Factoring + b + c Algebra 1 Essential Question How can you use algebra tiles to factor the trinomial b c into the product of two binomials? Factoring a Quadratic Trinomial ( + b + c) To factor a quadratic epression means to write it as the product of linear epressions. You can think of factoring trinomials of the form + b + c as reverse FOILing. ( + p)( + q) = ( + p)( + q) = In order to factor + b + c, you must find p and q such that: (p + q) = AND pq = 1. Factor the following (a) + + (b) - 8 + 1 Eample: + 6 + 8 = ( +? )( +? ) (c) + 11 + 10 (d) - 8 9 Solving Quadratic Equations by Factoring a. Rewrite b. Factor c. Use d. Solve Eample: = 10 (e) + 18 (f) - 17 + 60 (g) y y - 48 1
. Solve the Equation by Factoring (a) z + 11z = 6 (b) + 11 +18 = 0 (c) + 16 = -1. Factoring Trinomials with a G.C.F. Completely factor the trinomial. (a) 4 4 (b) 6 0 4 (c) 11 4 10 9 8 11 110 (d) 7 y 14 y 6 y 4. Solving Equations by Factoring Trinomials and a G.C.F. Solve each equation using the Zero Product Property. (a) 9 6 90 0 (b) y 1y 90y 0 (c) 4 4 6 9 (d) 7y 70y 147. The area of a right triangle is square miles. One leg of the triangle is 4 miles longer than the other leg. Find the length of each leg. 14
Name Date 7. Worksheet Factoring Common Monomials, Factoring Trinomials, and Solving Equations Factor the GCF for each polynomial. 1.. 1 7 6 1. 4 1 4 4. 1v 18v. 6 q 9 q 6. 0 6 6 Factor each trinomial of the form b c 7. 6 8 8. 4 9. 10. 8 11. 7 1 1. 6 1. 0 14. 8 16 1. 10 4 Factor the GCF first, and then factor the remaining trinomial. 16. 4 6 17. 4 8 1 18. 16 0 1
19. 4 0 16 0. 4 84 1. 4 4 4 64. 4 18. 9 108 4. 4 4 108 Solve the following equations by factoring.. 40 0 6. 16 6 0 7. 7 6 8. 0 9. 1 18 0 0. 4 1 4 1 16
Lesson 7.6(a) Factoring Quadratic Trinomials: a b c Algebra 1 Essential Question How can you use algebra tiles to factor the trinomial a + b + c into the product of two binomials? Warm-Up Eercise Factor each epression completely. (a) 6 40 (b) 1 10 Find each product using the distributive property. (c) 1 (d) 4 7 Factoring: The Bo Method When the leading coefficient of a trinomial is greater than 1, the factors of both the leading coefficient and the constant play a role in determining. The bo method helps us organize the work needed to factor each epression. Eample: This method will only work if the greatest common factor is factored out first. Step 1 8 Factoring: The Bo Method- Steps 1. Insert the first term of the trinomial into the upper left bo.. Insert the last term into the lower right bo.. Find the product of the leading coefficient and the constant term. Be sure to carry all negative signs (if necessary) 4. Find and list all factors of the product from step.. Find the pair of factors that sum to the middle term s coefficient. 6. Insert each factor into the empty boes as terms. 7. Find the GCF of each row and each column - If the front bo of each row or column is negative, then the GCF is negative - If there is nothing in common, then the GCF is 1 8. Write the GCF s from the rows as a binomial, and write the GCF s form the columns as a binomial. Step Step 17 Step 7
1. Factoring Using the Bo Method Factor each epression completely. (a) 9 1 4 (b) 10 8 (c) 1y 17y 6 Practice Problems - Complete all problems in the space below. Show all work Factor each epression completely. (a) b 11b 6 (b) 4 8 (c) (8m 10m ) 18
Worksheet 7.6(a) Factoring Quadratic Trinomials: a b c Name Factor each polynomial of the form a b c. Use the Bo method to factor. 1. 6 11. 9. 6 11 10 4. y 7y 44. 14y 1y 4 6. 6 70 7. 19 10 8. 7 1 1 9. 0 10. 11 4 11. 10 8 1. 7 6 1. 14. 0 1 1. 9 19
Lesson 7.6(b) Solving Quadratic Equations w/ Zero Product Property Algebra 1 Warm-up Eercises Factor the following epressions (a) 4 7 (b) 11 + + 1 (c) 1 11 + 1. Solving Quadratic Equations Using Zero Product Property Solve the equation Solving Quadratic Equations (a) 1 + 4 + = 0 1... (b) 7 + 14n + 1n = 6n + 11 (c) 10t + t = -11t t + 48 (d) + 8 = 8 1. Factoring Polynomials Involving the Greatest Common Factor Factor each epression completely. (a) 14 8 (b) 7n 6 4 (c) 1 4 6 10 0
. Completely Factoring Polynomials of Higher Degrees Factor each epression completely. (a) b 11b 6 (b) 4 8 (c) 40 m 0m 1 (d) 48 96 7 7.6C Worksheet: Factoring the GCF and a b c Name: In each of the polynomials, factor out the GCF and then factor the remaining trinomial. 1. 1 7 6. 4 10 8. 40 108 0 4. 1 10. 0 60 6. 1d d 4d 1
7. 1g g 4 6g 69 8. 8 6 4 14 1 9. 90 180 0 10. 16 8 48 11. 0 4 1 1. 40 0 0 1. 4 6 4 14. 4 0 1. 9 1
Solve the equations by factoring. 16. 6 + 1 + = 0 17. + 7 = - 18. 10 = = - 1 19. 1 + = - 0. 6 10 4 = 0 1. 6 7 + 7 = 0. 8 + 10 + = 0. 4 8 = 0
Lesson 7.7 Difference of Squares and Perfect Square Trinomials Algebra 1 Essential Question How can you recognize and factor special products? Warm-Up Eercise Find the product. (a) ( 1)( 1) (b) ( y 1)(y 1) (c) ( n 1) (d) ( n ) 1. Factoring the Difference of Two Squares Factor each epression. (a) 64 (b) 4 9 Difference of Two Squares Since the special product of two binomials has the following property: The difference of two squares reverses the process by factoring (c) 6 (d) 0a 16 11 y Perfect Square Trinomial Since the special product of two binomials has the following property: By reversing the process. Factoring Perfect Squares Trinomial Factor each epression and identify the pattern for each perfect square trinomial. (a) 6 9 (b) 6 4 4 (c) 10y 00y 1000 (d) 16y 4y 9. Solving Equations Involving Special Product Factorization Solve the equation. (a) 0 (b) 1 18 4
Lesson 7.8 Factoring a Polynomial Completely Algebra 1 Essential Question How can you factor a polynomial completely? 1 EXPLORATION: Matching Standard and Factored Forms Work with a partner. Match the standard form of the polynomial with the equivalent factored form on the net page. Eplain your strategy. a. d. b. 4 4 e. c. f. g. h. 4 i. j. k. l. 4 m. n. 4 4 o. A. B. C. D. E. F. G. H. I. J. K. L. M. N. O. Prime Polynomials Factoring a Polynomial Completely A polynomial is completely factored if it can be written as the product of monomials and prime polynomials. E.
1. Factor the Polynomials Completely. If it is prime, then say so. (a) 11 (b) 4 9 (c) 0 (d) 4 7 (e) 1. Factoring By Grouping Factor the polynomials (a) 4 6 4 (b) 6 7 Factoring By Grouping You can use the distributive property to factor some polynomials that have FOUR terms.. Solving Equations By Factoring Completely Find all solutions to each equation. (a) 16 100 0 (b) 0 (c ) 1 0 0 (d) y y 0 0 ( e ) c 81c 0 4 6
7 Unit 9 Test Review I. Identifying Polynomials Name each polynomial by its degree, number of terms, and leading coefficient 1.. 6. 7 4 II. Polynomial Operations Perform the indicated operation for each polynomial epression below. 4. 1 7. 4 1 6. 4 m m m m m 7. 4 7 7 8. 7 8 t t t t 9. 1 1 4 4 10. 6 11. 4 8 1 4 1. 4 7 1. 1 4 14. 1 8 6 4 1. 16. 4 7 17. 4 4 18. 11
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. 6. Look for a difference of two squares or a perfect 4 4 square trinomial.. Factor a trinomial of the form a b c into a product 1 of binomial factors. 4. Factor a polynomial with four terms by grouping. 4 4 1 4 I. Factoring Polynomials. Factor each epression completely 1. 4 t t 18. 4 w 1w. 1 10 4 64 4. 11 7 4 11. 1y y 10 9 6 1y 1 6. 1 7 7. 17 7 8. 0 100 9. 8 7 10. 44 1 11. 90 1. 10 0 8
1. y 10y 1y 14. d d 4 d 60 1. 7m 8m 1m 16. 1 17. 9 6 1 18. 4 7 19. 1 49 14 0. 4 64 10 1. 4 0 16. 9. 64 9 4. 00 16. 6 16 6. 6 48 16 7. 9 60 100 9
8. 4 4 9. 4 1 0. d 10d d 1 Solve each equation. Be sure to find all solutions. 1. 1 64 0. 8. 14 0 4. 11 10 0. 1 10 8 6. 6 1 7. 48 0 8. 4 0 9. d d 9d 18 0 0