Note: This unit can be used as needed (review or introductory) to practice operations on polynomials. Math Background Previously, you Identified monomials and their characteristics Applied the laws of exponents and explored exponential functions Identified and evaluated expressions involving exponents In this unit you will Write polynomials in standard form Add, subtract, and multiply polynomials Multiply binomials to obtain trinomials or special Multiply special products You can use the skills in this unit to Work with and solve practical applications such as area and free fall Write polynomials as products to develop methods of solving quadratics Overall Big Ideas Polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication. Essential Questions Why is it helpful to be able to change the forms of quadratic expressions? How is the arithmetic of polynomials similar to the arithmetic of integers or real numbers? What are some of the identifying characteristics of polynomials? Note: A file Algebra Unit 09 Practice X Patterns can be useful to prepare students to quickly find sum and product. Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 1 of 14
Skill: Add and subtract polynomials connecting the arithmetic and algebraic processes. SSE.1a Interpret the structure of expressions 1. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. APR.1 Perform arithmetic operations on polynomials 1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. SSE.3b Write expressions in equivalent forms to solve problems 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. F.IF.7a Analyze functions using different representations 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page of 14
Teacher Note: The sample questions should be used to prepare for instruction. Sample Questions 1. Which is equivalent to x y xy? 3 y 6xy 3 y xy 3 y 6x y 4 3 9x y. Under what operations is the system of polynomials NOT closed? addition subtraction multiplication division 3. Which expression is equivalent to 6 4 3 5 8 7 x x x x? x x 11x 14x 8 14x 11x 8 4. Subtract: 9 y 5 y 6 3 y y 4 6y 4y 6y 4y 10 6y 6y 6y 6y 10 Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 3 of 14
5. Expand the expression 7. 9x 4x 49 9x 4x 49 9x 49 9x 49 For questions 6-8, answer each with respect to the system of polynomials. 6. The system of polynomials is closed under subtraction. True False 7. The system of polynomials is closed under division. True False 8. The system of polynomials is closed under multiplication. True False Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 4 of 14
Notes: 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 expression which is the sum or difference of terms of the form integer and a is a real number Degree of a Term: the value of the exponent of the variable Degree of a Polynomial: the largest degree of its terms k ax, where k is a nonnegative 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 3 4 Name Constant Linear Quadratic Cubic Quartic By Number of Terms: # of Terms 1 3 4 and up Name Monomial Binomial Trinomial Polynomial Ex 1: Write the polynomial in standard form. State its degree and leading coefficient. Then classify by the degree and number of terms. 3 74x x Standard Form: Write the terms in descending order of the exponents. 3 4x x 7 Degree of the Polynomial: The exponent of the first term. Degree = 3 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 examples of various classifications, i.e. quadratic monomial, linear binomial, etc. Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 5 of 14
Operations with Polynomials Recall: Expanded notation of a number. 453 4 100 5 10 3 1 4 10 5 10 3 1 31 3 100 1 10 1 3 10 1 10 1 As polynomials, we can write these as 4x 5x3 1x When we add 453 + 31, we must line up the place values. This is just like adding like terms. 453 4 10 5 10 3 1 31 3 10 1 10 1 765 7 10 6 10 5 1 Ex : Add in your head 341 + 14 + 13. 4x 5x3 1x 7x 6x5 You have 300 + 00 + 100, that s 600, adding the tens, we have 40 + 10 + 30 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) Ex 3: Find the sum. Write the answer in standard form. 4x 3 x x 8 6 3 x Method 1 (Vertical Method): Write the polynomials in standard form and line up like terms. 3 4x x x 8 x 3 x 6 x 14 3 3 x x 14 Note: This is a cubic trinomial. Method (Horizontal Method): Find the like terms and add them. 4x 3 x x 8 6 3 x 4x 3 3 x x x 8 6 x 3 x 14 Subtraction: to subtract polynomials, add the opposite Ex 4: Find the difference. Write the answer in standard form. x 5 6 5x 4x Method 1 (Vertical Method): Write the polynomials in standard form and line up like terms. Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 6 of 14
trinomial x 5 4x 5x 6 add the opposite: x 5 4x 5x 6 8x 11 Note: This is a quadratic Method (Horizontal Method): Find the like terms and add them. Add the opposite. x x x x x x x x x x 3 5 6 5 4 4 3 5 5 6 3 8 11 Addition and Subtraction: Ex 5: Simplify the expression. Classify the remaining polynomial. x 3 1 x 3 8 Use the horizontal method. Add the opposite: x 3 1 x 3 8 Combine like terms: x x x 3 x 3 3 3 1 8 5 Classify: This is a constant monomial. Modeling with Polynomials Ex: 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 13t 79,589 M 1164t 75,6 To find the total population, we need to add the two polynomials above. 13 79589 1164 756 P t t P387t15511 To estimate the population in 009, we will use t 59, since 009 is 59 years after 1950. P 387 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 polynomial. 3a 3 4a 3 a 4 a 3 1.a a 9 QOD: Is it possible for a polynomial to be classified as a linear trinomial? Explain why or why not. Name two classifications for polynomials that are impossible. Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 7 of 14
Skill: multiply polynomials connecting the arithmetic and algebraic processes. APR.1 Perform arithmetic operations on polynomials 1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Multiplying Polynomials Recall: You have multiplied a monomial by a polynomial using the distributive property. Ex 6: Find the product. 4x x 1 Multiply each term of the trinomial by the monomial. 4x x 4x 4x 1 4x 1x 4x 4 3 Method 1: To multiply by a binomial, use the distributive property twice, then combine like terms. Ex 7: Find the product. 5x 8x 3 Distributive property: Combine like terms: x 8x 3 5 x 5 8x 5 3 3 6x 4x 9x 10x 40x 15 3 6x 14x 31x 15 Method : Multiply polynomials vertically. Recall: When multiplying whole numbers, we use a vertical method. 3 1 3 64 67 Ex 8: Find the product. x 1 x 1 multiply by 1 6 4 multiply by x x x 6x 7x add like terms Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 8 of 14
Method 3: 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. Ex 9: Find the product. 8x 1 First x Outer 1 Inner 8x Last 8 1 8x 8 5x 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! Ex 10: Multiply the polynomials vertically. 4y y 7y 8 4y y7 y 8 3 y 8y 56 multiply by 8 3 8 14 multiply by y y y y 3 8y 30y y 56 add like terms Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 9 of 14
Polynomials in Real Life Ex 11: A piece of paper has margins that are inches on the sides and 3 inches at the top and bottom. The height-to-width ratio of the usable part of the paper is 3: as shown in the diagram. Write a polynomial expression that represents the total area of the paper, including the margins. Find the area when x 10. Height of Paper = 6 Width of Paper = x 4 6 x 4 6x 1x 1x 4 6x 4x 4 Area of Paper = Height x Width = x : When 10 6 10 4 10 4 600 40 4 864 sq. in. You Try: Write a polynomial expression for the area of a trapezoid with bases 1 height of 4x 8. x and 4, and a QOD: List some advantages and disadvantages of each method of multiplying polynomials: horizontal, vertical, and FOIL. Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 10 of 14
Sample Practice Question(s): 9. Which expression represents the perimeter of the triangle shown below? x + 3 4x + 1 x x + 4 x x 5x 3 5x 6 8 10. Subtract the following polynomials: 4y 7 y 5 y 5y 3 y y y 1y 8 6y y 6y 1 y 8 11. Multiply the binomials x 3 1. 5x 7x 3 5x 5x 3 6x 7x 3 6x 5x 3 Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 11 of 14
1. Multiply the polynomials: x 5x 4 3 x 7x 11x 0 3 x 7x 19x 0 3 x 1 11x 0 3 x 1 19x 0 Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 1 of 14
Skill: multiply polynomials recognizing special products. Exploration: Find the products using FOIL. x x x x x 4 x 4 x x 1x 1 4x x x 1 4x 1 4x 1 5t 35t 3 5t 15t 15t 9 5t 9 5t 3 a ba b a ab ab b a b Special Pattern Sum and Difference Pattern a ba b a b Ex 1: Find the product without using FOIL. 3t43t 4 Use the sum and difference pattern. Exploration: Find the products using FOIL. 3t 4 3t 4 3t 4 9t 16 x 3 x 3 x 3 x 9 x 6x 9 x 5 x 5 x 5 x 5x 5x 5 x 10x 5 x 7 x 7 x 7 4x 14x 14x 49 4x 8x 49 a b a b a b a ab ab b a ab b Special Pattern Square of a Binomial Pattern a b a ab b a b a ab b Ex 13: Find the product without using FOIL. 8y 3 8y 3 8y 8y 3 3 64 y 48y 9 Use the square of a binomial pattern. Ex 14: Find the product without using FOIL. 4y 5 4x 5 4x 4x 5 5 16x 40x 5 Use the square of a binomial pattern. Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 13 of 14
Using Special Products to Multiply Numbers Ex 15: Multiply 3 8 using mental math. Rewrite the two factors. 3 30 and 8 30 30 30 30 900 4 896 Multiply using the sum and difference pattern. Ex 16: Multiply 5 using the square of a binomial pattern. Rewrite 5 as a sum (or difference). 5 0 5 0 5 0 0 5 5 400 00 5 65 Multiply using the square of a binomial pattern. You Try: Find the product of the following using special patterns. 3b73b7 1 5n QOD: Is the following statement true or false? If it is false, give a counterexample. a b a b Sample CCSD Common Exam Practice Question(s): Expand the expression 5. 9x 5 9x 5 9x 30x 5 9x 30x 5 NOTE: The following projects can be found on the website. They may be used to give students quick practice adding and subtracting numbers. This skill will be very useful for quadratics (add to get the middle term coefficient; multiply to get the last term coefficient, or constant. Sum and Product Practice X Factor Alg I Unit 9 Notes PolynomialsOperationsSpecialProducts 3/16/015 Page 14 of 14