Lesson 10.1 Polynomials Objectives Classify polynomials. Use algebra tiles to add polynomials. Add and subtract polynomials. A contractor is buying paint to cover the interior of two cubical storage tanks. Is there an algebraic epression for finding the amount of paint needed to cover both objects? The first storage tank has sides with a length of feet. The area of one of the faces of this tank is 2 square feet. Because the tank has si faces, the total interior area is 6 2 square feet. The second storage tank has sides of length y feet and an eterior area of 6y 2 square feet. The total interior area of the two tanks is described by the epression 6 2 6y 2. Monomials, Binomials, and Trinomials The epression 6 2 6y 2 contains two monomial terms. A monomial is a constant, a variable, or the product of a constant and one or more variables. The eponent of the variable in a monomial is the degree of that monomial. The degree of 7 is 1, the degree of 4 3 is 3, and the degree of 5 is 0. The sum of one or more monomials is a polynomial. For eample, 3 2 4 2 is a polynomial in one variable. The monomial terms are 3 2, 4, and 2. The degree of a polynomial is the greatest degree of all of the terms in the polynomial. The degree of 3 2 1 4 2 is 2 and the degree of 2 2 2 4 4 1 is 4. In this chapter, you will discover patterns that can be represented by three special polynomials. Name Number of Terms Eample Monomial 1 3 2 Binomial 2 3 2 4 Trinomial 3 3 2 3 2 The epression 3 2 is a monomial because it has one term. The epression 3 2 4 is a binomial because it has two terms. The epression 3 2 3 2 is a trinomial because it has three terms. y y y 560 Chapter 10 Polynomials and Factors
Algebra tiles can model monomials, binomials, and trinomials. Previously, you used algebra tiles to model an epression such as 3 2. To model 2, a new algebra tile is needed. Use a tile that is units on each side. The area of the tile is represented by 2. 1 Area 5 1 Area 5 5 Area 5 2 Algebra tiles model for Algebra tiles model for 2 3 2 2 3 2 Adding and Subtracting Polynomials You can use algebra tiles or like terms to add and subtract polynomials. Activity Adding Trinomials 1 Use algebra tiles to model 3 2 4 1. 2 Use different algebra tiles to model 2 2 3 2. 3 Combine the algebra tiles. 4 Record your result. 5 What is the sum of (3 2 4 1) and (2 2 3 2)? 5 2 3 6 Complete the equation. (3 2 4 1) (2 2 3 2)? 5 2 3 10.1 Polynomials 561
Recall that monomials that have the same variable factors are like terms. For eample, 3 2 and 2 2 are like terms because each term has 2 as a common factor. You can add like terms using the Distributive Property. 3 2 2 2 (3 2) 2 5 2 If you add the coefficients of the like terms in the epression the result is 3 2 6. (2 2 3 4) ( 2 3 2) Critical Thinking Eplain how to simplify the following epressions. What is the degree of the resulting polynomial? (2 2 3 1) ( 2 3 2) Add the coefficients of like terms. The result is 3 2 3. The degree is 2. Eample 1 Subtracting Polynomials Subtract. (3 2 5) ( 2 2 3) Solution You can find the difference by using the basic properties you learned in earlier chapters. Horizontal Method (3 2 5) ( 2 2 3) (3 2 5) ( 1)( 2 2 3) Definition of subtraction Vertical Method 5 3 2 5 ( 2 2 3) Distributive Property 5 3 2 2 2 5 3 Rearrange terms. 5 2 2 2 2 Add like terms. 3 2 0 5 ( 2 2 3) Use zero as coefficient of missing term. Subtract like terms. 2 2 2 2 562 Chapter 10 Polynomials and Factors
Ongoing Assessment Use any method to simplify ( 4 2 ) (2 2 3 5). 6 2 4 5 Eample 2 The Perimeter of a Triangle Find the simplest epression for the perimeter of triangle ABC. What is the degree of the epression for the perimeter? A 3a 4 2a + 7 B 4a C Solution The perimeter is the total distance around the triangle. To find the perimeter, add the lengths of the three sides; that is, simplify If you rearrange the terms, the result is 3a 2a 4a 4 7. Now combine like terms. The perimeter is 9a 3. The degree of the epression is 1 because 1 is the greatest degree of the two terms. (3a 4) (2a 7) 4a. One way to check the solution is to substitute a number for a in both the original epression and in the simplified epression. In this case, substitute 10 for a to see if the results are equal. 3a 4 2a 7 4a 9a 3 [3(10) 4] [2(10) 7] 4(10) 9(10) 3 26 27 40 90 3 93 5 93 4 Because the results are equal, the simplification is correct. 10.1 Polynomials 563
Lesson Assessment Think and Discuss 1. Eplain what is meant by like terms of an epression. 2. Eplain the meaning of monomial, binomial, and trinomial. Give an eample of each. 3. If polynomial means many names, eplain what polygon means. 4. How is the Distributive Property used to combine like terms? 5. Eplain how to determine the degree of a polynomial. Practice and Problem Solving Classify each of the following epressions as a monomial, binomial, or trinomial. State the degree of the polynomial. 6. 4 2 3 binomial; 2 7. 15 4 monomial; 4 8. 5 binomial; 1 9. 2 monomial; 0 10. 7 2 3 1 trinomial; 2 11. 3 monomial; 1 Use algebra tiles to find each sum. 12. (3 5) (2 3) 5 8 13. (4 2 3 7) (2 2 5 3) 6 2 8 10 14. (3 2 5 2) (4 5) 3 2 3 15. (7 5) (3 2) 10 7 16. (2 2 3 6) ( 2 2 4) 2 2 Add. 17. (2 7) ( 2 3 5) 2 5 12 18. (3) (2 2 3 4) 2 2 4 19. (3 2 7 4) ( 2 7 4) 4 2 20. (4 2 5 7) (3 2 7) 7 2 5 21. ( 2 10 8) (6 2 10 13) 7 2 5 22. ( 3 2 5 12) ( 2 6 3) 2 2 9 23. (5 2 15 1) ( 7 2 14) 2 2 15 15 564 Chapter 10 Polynomials and Factors
Subtract. 24. (2 5) ( 2) 3 25. (5 4) (2 7) 3 3 26. (3 2 8 5) (2 2 1) 2 7 4 27. (6 2 5 7) (3 3) 6 2 2 10 28. (4 2 3 4) (5 4) 4 2 8 29. ( 2 5 3) (3 2 7 4) 2 2 2 7 30. ( 4 2 3 10) (2 2 9 6) 6 2 6 4 31. The sides of a triangle are represented by the epressions 2 2 3, 4 2 7, and 5 4. Write the simplest epression for the perimeter of the triangle. 6 2 2 2 7 32. The length of a rectangle is represented by the epression 3 2 2 1. Its width is represented by the epression 2 2 3. Write the simplest epression for the perimeter of the rectangle. 8 2 4 Mied Review 33. Solve the distance formula d rt for the rate r. 34. Find the area of a circular garden with radius 5 meters. about 78.5 square meters Solve each equation. 35. 2 8 10 9 36. 13 45 8 4 37. 3 14 4 2 38. If the perimeter of a square is 40 cm, what is the length of a diagonal across the square? about 14.14 centimeters 39. Barbara bought several CDs for $12 each and paid $5 for a poster. If she spent a total of $41, how many CDs did she buy? 3 10.1 Polynomials 565