Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. Teacher Note: A nice way to illustrate operations with polynomials is by using algebra tiles. If you have them available, use them with your students throughout this unit. Polynomial: an epression which is the sum or difference of terms of the form nonnegative integer and a is a real number Degree of a Term: the value of the eponent of the variable Degree of a Polynomial: the largest degree of its terms k a, where k is a Standard Form of a Polynomial: the terms of the polynomial are written in descending order, from largest degree to smallest degree Leading Coefficient: the coefficient of the first term of a polynomial when it is written in standard form Classifying Polynomials By Degree: Degree 0 1 4 Name Constant Linear Quadratic Cubic Quartic By Number of Terms: # of Terms 1 4 and up Name Monomial Binomial Trinomial Polynomial E: Write the polynomial in standard form. State its degree and leading coefficient. Then classify by the degree and number of terms. 7 4 Standard Form: Write the terms in descending order of the eponents. 4 7 Degree of the Polynomial: The eponent of the first term. Degree = Leading Coefficient: The coefficient of the first term. Leading Coefficient = 4 Classify: By degree, it is a cubic. By the number of terms, it is a trinomial. Cubic Trinomial On Your Own: Have students write their own eamples of various classifications, i.e. quadratic monomial, linear binomial, etc. Page 1 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Operations with Polynomials Recall: Epanded notation of a number. 45 4 100 5 10 1 4 10 5 10 1 1 100 1 10 1 10 1 10 1 As polynomials, we can write these as 4 5 1 When we add 45 + 1, we must line up the place values. This is just like adding like terms. 45 4 10 5 10 1 1 10 1 10 1 765 7 10 6 10 5 1 E: Add in your head 41 + 14 + 1. 4 5 1 7 6 5 You have 00 + 00 + 100, that s 600, adding the tens, we have 40 + 10 + 0 which is 80, and finally adding 1 + 4 + or 7, the sum is 687. Addition: to add polynomials, add like terms (terms with the same variable part) E: Find the sum. Write the answer in standard form. 4 8 6 Method 1 (Vertical Method): Write the polynomials in standard form and line up like terms. 4 8 6 14 14 Note: This is a cubic trinomial. Method (Horizontal Method): Find the like terms and add them. 4 8 6 4 86 14 Subtraction: to subtract polynomials, add the opposite E: Find the difference. Write the answer in standard form. 5 65 4 Method 1 (Vertical Method): Write the polynomials in standard form and line up like terms. Page of 19 McDougal Littell: 10.1 10., 10.5 10.8
5 4 5 6 add the opposite: 5 4 5 6 8 11 Note: This is a quadratic trinomial Method (Horizontal Method): Find the like terms and add them. Add the opposite. 5 65 4 4 5 56 8 11 Addition and Subtraction: E: Simplify the epression. Classify the remaining polynomial. 1 8 Use the horizontal method. Add the opposite: 1 8 Combine like terms: 18 5 Classify: This is a constant monomial. Modeling with Polynomials E: Projected from 1950 through 010, the female population F and the male population M of the United States (in thousands) can be modeled by the following equations, where t is the number of years since 1950. Find a model that represents the total population P of the U.S. from 1950 through 010. Use the model to estimate the total population of the U.S. in 009. F 1t 79,589 M 1164t 75,6 To find the total population, we need to add the two polynomials above. P 1t79589 1164t756 P 87t15511 To estimate the population in 009, we will use t 59, since 009 is 59 years after 1950. P 87 59 15511 96,044 thousand, or 96,044,000 people You Try: Add or subtract the polynomials. Write your answer in standard form and classify the a 4a a 4 a 1.a a 9 polynomial. QOD: Is it possible for a polynomial to be classified as a linear trinomial? Eplain why or why not. Name two classifications for polynomials that are impossible. Page of 19 McDougal Littell: 10.1 10., 10.5 10.8
Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. Multiplying Polynomials Recall: You have multiplied a monomial by a polynomial using the distributive property. E: Find the product. 4 1 Multiply each term of the trinomial by the monomial. 4 4 4 1 4 1 4 4 Method 1: To multiply by a binomial, use the distributive property twice, then combine like terms. E: Find the product. 5 8 Distributive property: Combine like terms: 8 5 5 8 5 6 4 9 10 40 15 6 14 1 15 Method : Multiply polynomials vertically. Recall: When multiplying whole numbers, we use a vertical method. 1 64 67 E: Find the product. 1 1 multiply by 1 6 4 multiply by 6 7 add like terms Method : FOIL This method can be used only when multiplying two binomials. This acronym comes from the distributive property in the order of First Outer Inner Last. Page 4 of 19 McDougal Littell: 10.1 10., 10.5 10.8
E: Find the product. 8 1 First Outer 1 Inner 8 Last 8 1 88 5 8 Teacher Note: If you draw arcs over each product (arcs above for First and Last, arcs below for Outer and Inner) it will make the smiling man. Try it the kids will love it! E: Multiply the polynomials vertically. 4y y7y 8 4y y 7 y 8 y 8y 56 multiply by 8 y y y y 8 14 multiply by y y y 8 0 56 add like terms Polynomials in Real Life E: A piece of paper has margins that are inches on the sides and inches at the top and bottom. The height-to-width ratio of the usable part of the paper is : as shown in the diagram. Write a polynomial epression that represents the total area of the paper, including the margins. Find the area when 10. Height of Paper = 6 Width of Paper = 4 Area of Paper = Height Width = 6 4 6 114 6 4 4 : When 10 6 10 4 10 4 600 40 4 864 sq. in. You Try: Write a polynomial epression for the area of a trapezoid with bases 1 and a height of 4 8. and 4, QOD: List some advantages and disadvantages of each method of multiplying polynomials: horizontal, vertical, and FOIL. Page 5 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Sample CCSD Common Eam Practice Question(s): 1. Which epression represents the perimeter of the triangle shown below? + 4 + 1 + 4 A. B. C. D. 5 5 6 8. Subtract the following polynomials: 4y 7y5y 5y A. B. C. D. y y y 1y 8 6y y 6y 1y 8. Multiply the binomials 1. A. B. C. D. 5 7 5 5 6 7 6 5 Page 6 of 19 McDougal Littell: 10.1 10., 10.5 10.8
4. Multiply the polynomials: 5 4 A. B. C. D. 7 11 0 7 19 0 1 11 0 1 19 0 Page 7 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. (special products) Eploration: Find the products using FOIL. 4 4 114 14 14 1 5t5t5t 15t15t9 5t 9 5t ababa ababb a b Special Pattern Sum and Difference Pattern a bab a b E: Find the product without using FOIL. t 4t 4 Use the sum and difference pattern. Eploration: Find the products using FOIL. t4 t4 t 4 9t 16 9 69 5 5 5 555 105 7 7 7 4 1414494 849 ab ab ab a ababb a abb Special Pattern Square of a Binomial Pattern a b a abb a b a abb E: Find the product without using FOIL. 8y 8y 8y 8y 64y 48y 9 Use the square of a binomial pattern. E: Find the product without using FOIL. 4y 5 45 4 4 5 5 16 40 5 Use the square of a binomial pattern. Using Special Products to Multiply Numbers E: Multiply 8 using mental math. Page 8 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Rewrite the two factors. 0 and 8 0 0 0 0 900 4 896 Multiply using the sum and difference pattern. E: Multiply 5 using the square of a binomial pattern. Rewrite 5 as a sum (or difference). 5 0 5 Multiply using the square of a binomial pattern. 0 5 0 0 5 5 400 00 5 65 You Try: Find the product of the following using special patterns. b 7b 7 1 5n QOD: Is the following statement true or false? If it is false, give a countereample. ab a b Sample CCSD Common Eam Practice Question(s): Epand the epression 5. A. B. C. D. 9 5 9 5 9 0 5 9 0 5 Page 9 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. (distributive property) Review: Greatest Common Factor (GCF) E: Find the GCF of 45 and 60. Recall: The GCF is the largest factor that the numbers have in common. We can write the prime factorization of each: 45 5 60 5 Now write all of the factors they have in common: 5 15 The GCF is 15. E: Find the GCF of 6 y and 16. Write out each term as a product of factors: 6 y y 16 Write all of the factors they have in common: Factoring Using the Distributive Property the polynomial 8. Step One: Find the GCF of the terms. 8 GCF = Step Two: Use the distributive property to factor the GCF out of the polynomial. 4 the polynomial 14 y1 y 7 y. Step One: Find the GCF of the terms. 14 7 y y y yy 1 7 y y 7 7 GCF = 7 y 7yy1 Step Two: Use the distributive property to factor the GCF out of the polynomial. 7yy1 Note: You can check your answers by multiplying using the distributive property. Page 10 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Factoring by Grouping the polynomial 6. Step One: Group the first two terms and last two terms. 6 Step Two: Factor the GCF from both sets of terms. Step Three: Factor the common factor using the distributive property. the polynomial n n n 6 18. Step One: Group the first two terms and last two terms. n 6n n 18 Step Two: Factor the GCF from both sets of terms. n n 6n 6 Step Three: Factor the common factor using the distributive property. n 6n You Try: Find the GCF and factor it out of the polynomial. 18 6y 4y 6 4 QOD: How can you find the GCF of a variable epression without writing out all of the variables as factors? Page 11 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. Factoring a Quadratic Polynomial Multiply p q using FOIL. To factor b c b q p, and c pq. into two binomials, p q q p pq q p pq, we must find values for p and q such that 5 4. Step One: Find values for p and q such that pq 4 and p q 5. We will list all of the factors of 4: 4 1414 The two factors that have a sum of 5 are 1 and 4, so p 1 and q 4. Step Two: Write each factor p q 1 4 Note: Because multiplication is commutative, we could also write our answer as 4 1 Check your answer using FOIL! 10 4. Step One: Find values for p and q such that pq 4 and p q 10. 4 14 1 8 4 6 We will list all of the factors of 4: 14 1 846 The two factors that have a sum of 10 are 4 and 6, so p 4 and q 6. Step Two: Write each factor p q 4 6 Check your answer using FOIL! 8 9. Step One: Find values for p and q such that pq 9 and p q 8. We will list all of the factors of 9: 91991 The two factors that have a sum of 8 are 9 and 1, so p 9 and q 1. Step Two: Write each factor p q 9 1 Check your answer using FOIL! Page 1 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Factoring Quadratic Trinomials in the Form a b c a, 1 the trinomial 11 5. Step One: Multiply ac. Find values for p and q such that pq ac, and p q b ac 10 110 5 110 5 The two factors that have a sum of 11 are 1 and 10. Step Two: Split the middle term (b) into two terms p Step Three: Factor by grouping. Check your answer using FOIL! q. 110 5 1 105 1 5 1 1 5 6n 11n. Step One: Multiply ac. Find values for p and q such that pq ac, and p q b ac 1 11 11 664 4 The two factors that have a sum of 11 are 1 and 1. Step Two: Split the middle term (b) into two terms p Step Three: Factor by grouping. Check your answer using FOIL! q. n n n 6 1 1 6n 1n 1n 6 1 6 1 n n n 6n 1n t t. 4 9 5 Step One: Multiply ac. Find values for p and q such that pq ac, and p q b ac 0 10 10 10 10 4 5 4 5 The two factors that have a sum of 9 are 4 and 5. Step Two: Split the middle term (b) into two terms p q. t t t 4 4 5 5 Step Three: Factor by grouping. 4t 4t 5t5 t 4t t1 5 1 t 14t 5 Check your answer using FOIL! Page 1 of 19 McDougal Littell: 10.1 10., 10.5 10.8
the trinomial 6 19 10. Step One: Multiply ac. Find values for p and q such that pq ac, and p q b ac 60 160 0 0 4 15 The two factors that have a sum of 19 are 4 and 15. Note: Because 60 has so many factors, we did not list all of them. b is positive, so we only need to list the positive factors and stop when we find the two that add up to 19. Step Two: Split the middle term (b) into two terms p Step Three: Factor by grouping. Check your answer using FOIL! q. 6 4 15 10 6 4 1510 5 5 You Try: Factor the following trinomials. 1. m 6m 5. 6y y 8. 1 8 4 QOD: Are all quadratic trinomials factorable? If not, write a trinomial that cannot be factored. Sample CCSD Common Eam Practice Question(s): Which of the following is a factor of? 5 1 A. 5 B. 5 C. 5 4 D. 5 6 Sample Nevada High School Proficiency Eam Questions (taken from 009 released version H): Factor: A 6 4 B 4 6 C 1 D 1 10 4 Page 14 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. (special products) Recall: Sum and Difference Pattern a bab a b a b Square of a Binomial Pattern: a abb a b a abb Previously we learned to multiply using the special products patterns. Now we will factor from these special products. Special Factoring Patterns Difference of Two Perfect Squares: a b aba b Perfect Square Trinomial: a abb ab a abb ab 9. Recognize this as the difference of two perfect squares. 9 Factor using the special factoring pattern. Check your answer using FOIL! 16 5y. Recognize this as the difference of two perfect squares. 16 5y 4 5y Factor using the special factoring pattern. 45y4 5y Check your answer using FOIL! y 6y 9. Recognize this as a perfect square trinomial. y 6y9y y y Factor using the special factoring pattern. Page 15 of 19 McDougal Littell: 10.1 10., 10.5 10.8
9 1 4. Recognize this as a perfect square trinomial. 9 14 Factor using the special factoring pattern. 9 16 f. 9 9 16 16 4 Recognize this as the difference of two perfect squares. f f f Factor using the special factoring pattern. f f 4 4 4y 0y 5. Recognize this as a perfect square trinomial. 4y 0y5 y y5 5 y 5 Factor using the special factoring pattern. You Try: Factor the following. 1. 49n 5m. 49 56 16 QOD: Describe how to check your answers when factoring. Page 16 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. (factoring completely) Factoring Completely: putting it all together Step One: Factor the GCF using the distributive property. **IMPORTANT!** Step Two: Factor the polynomial that remains using the ac method or special products. completely. 4 6 Step One: Factor the GCF, which is 4. 4 9 Step Two: The remaining polynomial is a difference of two perfect squares. Use the special factoring pattern. 4 completely. 10 1 n n n Step One: Factor the GCF, which is n. nn 5n 6 Step Two: Factor the remaining trinomial. nn6n 1 4 0y 58y 4 y. Step One: Factor the GCF, which is y. y 15y 9y 1 Step Two: Factor the remaining trinomial. y 5yy 4 Note: Because of the large numbers for a and c in the trinomial, a trial and error approach may work best for factoring. 15 1 60 completely. Step One: Factor the GCF, which is. 5 4 0 Step Two: Factor the remaining polynomial by grouping. 5 4 0 5 4 5 5 4 Page 17 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Step Three: The underlined factor is a difference of two perfect squares. Use the special factoring pattern to factor completely. 5 5 4 18. Step One: Factor the GCF, which is. 16 64 Step Two: The remaining polynomial is a perfect square trinomial. Use the special factoring pattern. 8 Quadratic Trinomials: Factorable or Not Factorable? Discriminant: In the quadratic trinomial a b c, the discriminant is equal to If the trinomial can be factored, the discriminant must be a perfect square. b 4ac E: Is the trinomial factorable? If yes, factor it. 5 11 Find the discriminant. a 5, b 11, c b 4ac 11 4 5 1140 81 81 is a perfect square, so the trinomial is factorable. Use the ac method. ac 10 1 10 5 1 10 5 1 5 1 5 1 E: Is the trinomial factorable? If yes, factor it. r 5r 1 Find the discriminant. a, b 5, c 1 b 4ac 5 4 1 51 7 7 is not a perfect square, so the trinomial is not factorable. Application Problem E: The volume of a bo can be modeled by the polynomial 4 1. Write three linear binomials that could represent the length, width, and height of the bo. The volume of the bo was found by multiplying the length, width, and height. To find the binomials, we will factor the epression that represents the volume. Page 18 of 19 McDougal Littell: 10.1 10., 10.5 10.8
Factor by grouping. 41 4 4 The underlined factor is a difference of two perfect squares. Use the special pattern to factor. The length, width, and height of the bo are,, and You Try: List all of the factors of b b b. 4 6 9 QOD: Why is it important to factor the GCF as the first step in factoring a polynomial? Sample CCSD Common Eam Practice Question(s): The area of a rectangular tabletop is represented by 5 4. Which pair of epressions could represent the dimensions of the tabletop? A. 6, 4 B. 8, C. 1, D. 4, 1 Page 19 of 19 McDougal Littell: 10.1 10., 10.5 10.8