# LESSON 6.2 POLYNOMIAL OPERATIONS I

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1 LESSON 6.2 POLYNOMIAL OPERATIONS I Overview In business, people use algebra everyday to find unknown quantities. For example, a manufacturer may use algebra to determine a product s selling price in order to maximize the company s profit. A landscape architect may be interested in finding a formula for the area of a patio deck. To find these quantities, you need to be able to add, subtract, multiply, and divide polynomials. Explain Concept 1 has sections on Definitions Degree of a Polynomial Writing Terms in Descending Order Evaluating a Polynomial Adding Polynomials Subtracting Polynomials CONCEPT 1: ADDING AND SUBTRACTING POLYNOMIALS Definitions A monomial is an algebraic expression that contains exactly one term. The term may be a constant, or the product of a constant and one or more variables. The exponent of any variable must be a nonnegative integer (that is, a whole number). The following are monomials: 12 x 2 5wy gt2 4.35T A monomial in one variable, x, can be written in the form ax r, where a is any real number and r is a nonnegative integer. LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 383

2 A polynomial with 4 terms is called a four term polynomial. A polynomial with 5 terms is called a five term polynomial, and so on. The following are not monomials: x 2 3 The denominator contains a variable with a positive exponent. So the term cannot be written in the form ax r where r is a nonnegative integer. 3 x 2 There is a squared variable under a cube root symbol. So the term cannot be written in the form ax r where r is a nonnegative integer. A polynomial is the sum of one or more monomials. Here are some examples: 2x 3 5x 2 x 2 5 3xy 2 7x 5y 1 A polynomial with one, two, or three terms has a special name. Number Name of terms Examples monomial 1 x, 5y, 3xy 3, 5 binomial 2 x 1, 2x 2 3, 5xy 3 4x 3 y 2 trinomial 3 x 2 2x 1, 3x 2 y 3 xy 5 Example Determine if each expression is a polynomial. 24 x 2 a. 4w 3 b. 3x c. x 3 2 d. x 2 Remember: x 0 1, for x 0 x 1 x Solution a. The expression is a polynomial. It has one term, so it is a monomial. The term has the form aw r, where a 4 and r 3. b. The expression is not a polynomial. The term 2 4 x2 cannot be written in the form ax r where r is a nonnegative integer. c. The expression is not a polynomial. The term x 3 cannot be written in the form ax r where r is a nonnegative integer. d. The expression is a polynomial. It has two terms, so it is a binomial. Each term can be written in the form ax r : x 2 1x 1 2x TOPIC 6 EXPONENTS AND POLYNOMIALS

3 Degree of a Polynomial The degree of a term of a polynomial is the sum of the exponents of the variables in that term. For example, consider this trinomial: 6x 3 y 2 xy x 4. The degree of the first term is 5. 6x 3 y 2 degree The degree of the second term is 3. xy 2 x 1 y 2 degree The degree of the last term is x 4 In 3 5, the exponent does not contribute degree 4 to the degree because the base, 3, is not a variable. The degree of a polynomial is equal to the degree of the term with the highest degree. In this polynomial, the term with degree 5 degree 3 degree 4 the highest degree has degree 5. So this polynomial has degree 5. 6x 3 y 2 x 1 y x 4 The polynomial has degree 5. Writing Terms in Descending Order The terms of a polynomial in one variable are usually arranged by degree, in descending order, when read from left to right. For example, this polynomial x 3 7x 2 4x 2 contains one variable, x. The terms of the polynomial x 3 7x 2 4x 1 2x 0 are arranged by degree in descending order. degree degree degree degree LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 385

4 Example Arrange the terms of this polynomial in descending order and determine the degree of the polynomial: 7x 3 8 2x x 4 Solution Write 8 as 8x 0. Write 2x as 2x 1. 7x 3 8x 0 2x 1 x 4 Arrange the terms by degree in x 4 7x 3 2x 1 8x 0 descending order (4, 3, 1, 0). The last two terms may be written x 4 7x 3 2x 8 without exponents. The term of highest degree is x 4. The degree of x 4 is 4. So, the degree of this polynomial is 4. Evaluating a Polynomial To evaluate a polynomial, we replace each variable with the given number, then simplify. Example Evaluate this polynomial when w 3 and y 2: 6w 2 4wy y 4 5 Solution Substitute 3 for w and 2 for y. 6(3) 2 4(3)(2) (2) 4 5 First, do the calculations with 6(9) 4(3)(2) 16 5 the exponents. Multiply Add and subtract. 19 Adding Polynomials To add polynomials, combine like terms. Recall that like terms are terms that have the same variables raised to the same power. That is, like terms have the same variables with the same exponents. Like terms 3x, 12x 8xy 2, 5.6xy 2 24, 11 4xy, 6yx NOT like terms 7x, 5xy The variables do not match. 3x 2 y 3, 2x 3 y 2 The powers of x do not match. The powers of y do not match. 386 TOPIC 6 EXPONENTS AND POLYNOMIALS

5 Example Find: (5x 3 13x 2 7) (16x 3 8x 2 x 15) Solution (5x 3 13x 2 7) (16x 3 8x 2 x 15) Remove the parentheses. 5x 3 13x x 3 8x 2 x 15 Write like terms next to 5x 3 16x 3 13x 2 8x 2 x 7 15 each other. Combine like terms. 21x 3 5x 2 x 8 We can also place one polynomial beneath the other and add like terms. 5x 3 13x x 3 8x 2 x 15 21x 3 5x 2 x 8 Example Find the sum of (3z 3 2zy 2 6y 3 ) and (15z 3 5zy 2 4z 2 ). Solution Write the sum. (3z 3 2zy 2 6y 3 ) (15z 3 5zy 2 4z 2 ) Remove the parentheses. 3z 3 2zy 2 6y 3 15z 3 5zy 2 4z 2 Write like terms next 3z 3 15z 3 2zy 2 5zy 2 6y 3 4z 2 to each other. We can also place one polynomial beneath the other and add like terms. 3z 3 2zy 2 6y 3 15z 3 5zy 2 4z 2 18z 3 3zy 2 6y 3 4z 2 Combine like terms. 18z 3 3zy 2 6y 3 4z 2 Subtracting Polynomials To subtract one polynomial from another, add the first polynomial to the opposite of the polynomial being subtracted. To find the opposite of a polynomial, multiply each term by 1. For example: The opposite of 5x 2 is 5x 2. The opposite of 2x 7 is 2x 7. Here s a way to find the opposite of a polynomial: Change the sign of each term. Example Find: (18w 2 w 32) (40 13w 2 ) Solution (18w 2 w 32) (40 13w 2 ) Change the subtraction to (18w 2 w 32) (1)(40 13w 2 ) addition of the opposite. Remove the parentheses. 18w 2 w w 2 Write like terms next to 18w 2 13w 2 w each other. Combine like terms. 5w 2 w 72 So, (18w 2 w 32) (40 13w 2 ) 5w 2 w 72. We can also place one polynomial beneath the other and subtract like terms. 18w 2 w 32 (13w 2 40) To do the subtraction, we change the sign of each term being subtracted, then add. 18w 2 w 32 (13w 2 w 40) 5w 2 w 72 LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 387

6 Example Subtract (15z 2 5yz 2 4y 3 ) from (6y 3 10z 3 2yz 2 ). We can also place one polynomial beneath the other and subtract like terms. 6y 3 10z 3 2yz 2 (4y 3 5yz 2 15z 2 ) To do the subtraction, we change the sign of each term being subtracted, then add. 6y 3 10z 3 02yz 2 (4y 3 05yz 2 15z 2 ) 2y 3 10z 3 10yz 2 15z 2 Solution Be careful! Subtract A from B means B A. The order is important. Write the difference. (6y 3 10z 3 2yz 2 ) (15z 2 5yz 2 4y 3 ) Change the subtraction to addition of the opposite. (6y 3 10z 3 2yz 2 ) (1)(15z 2 5yz 2 4y 3 ) Remove the parentheses. 6y 3 10z 3 2yz 2 15z 2 5yz 2 4y 3 Write like terms next to each other. 6y 3 4y 3 10z 3 2yz 2 5yz 2 15z 2 Combine like terms. 2y 3 10z 3 7yz 2 15z 2 Here is a summary of this concept from Interactive Mathematics. 388 TOPIC 6 EXPONENTS AND POLYNOMIALS

7 CONCEPT 2: MULTIPLYING AND DIVIDING POLYNOMIALS Multiplying a Monomial By a Monomial To find the product of two monomials, multiply the coefficients. Then, use the Multiplication Property of Exponents to combine variable factors that have the same base. Example Concept 2 has sections on Multiplying a Monomial by a Monomial Multiplying a Polynomial by a Monomial Dividing a Monomial by a Monomial Dividing a Polynomial by a Monomial Find: 7m 3 n 4 6mn 2 Solution Write the coefficients next to each other. 7m 3 n 4 6mn 2 Write the factors with base m next to each other, and write the factors (7 6)(m 3 m 1 )(n 4 n 2 ) with base n next to each other. Use the Multiplication Property of (7 6)(m 3 1 n 4 2 ) Exponents. Multiplication Property of Exponents: x m x n x m n Simplify. 42m 4 n 6 Example Find: 1 3 w3 x 7 y 6w 2 y 5 Solution Write the coefficients next to each other. 1 3 w3 x 7 y 6w 2 y 5 Write the factors with base w next to each other, and write the factors (w 3 w 2 )(x 7 )(y 1 y 5 ) with base y next to each other. Use the Multiplication Property (w 3 2 )(x 7 )(y 1 5 ) of Exponents. Simplify. 2w 5 x 7 y 6 LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 389

8 Example Find: (5x 3 y)(3x 5 )(2xy 5 ) Solution Write the coefficients next to (5x 3 y)(3x 5 )(2xy 5 ) each other. Write the factors with base x next to each other and write (5 3 2)(x 3 x 5 x 1 )(y 1 y 5 ) the factors with base y next to each other. Use the Multiplication Property (5 3 2)(x )(y 1 5 ) of Exponents. Simplify. 30x 9 y 6 Multiplying a Polynomial By a Monomial To multiply a monomial by a polynomial with more than one term, use the Distributive Property to distribute the monomial to each term in the polynomial. Example Find: 8w 3 y(4w 2 y 5 w 4 ) Solution 8w 3 y(4w 2 y 5 w 4 ) Multiply each term in the polynomial by the (8w 3 y)(4w 2 y 5 ) (8w 3 y)(w 4 ) monomial, 8w 3 y. Within each term, write the coefficients next to each other. Write the factors with base w next to each other and write the factors with base y next to each other. (8 4)(w 3 w 2 )(y y 5 ) (8)(w 3 w 4 )(y) Use the Multiplication (8 4)(w 3 2 y 1 5 ) (8)(w 3 4 y) Property of Exponents. Simplify. 32w 5 y 6 8w 7 y 390 TOPIC 6 EXPONENTS AND POLYNOMIALS

9 Example Find: 5x 4 (3x 2 y 2 2xy 2 x 3 y) Solution 5x 4 (3x 2 y 2 2xy 2 x 3 y) Multiply each term in the polynomial by the monomial, 5x 4. (5x 4 )(3x 2 y 2 ) (5x 4 )(2xy 2 ) (5x 4 )(x 3 y) Within each term, write the coefficients next to each other. Write the factors with base x next to each other and write the factors with base y next to each other. (5 3)(x 4 x 2 y 2 ) (5 2)(x 4 x 1 y 2 ) (5 1)(x 4 x 3 y) Use the Multiplication Property of Exponents. (5 3)(x 4 2 y 2 ) (5 2)(x 4 1 y 2 ) (5 1)(x 4 3 y) Simplify. 15x 6 y 2 10x 5 y 2 5x 7 y Dividing a Monomial By a Monomial To divide a monomial by a monomial, use the Division Property of Exponents. (Assume that any variable in the denominator is not equal to zero.) Division Property of Exponents x m xn x m n for m n and x 0 x m 1 xn x n m for m n and x 0 Example Find: 36w 5 xy 3 9w 2 y 7 Solution 36w 5 xy 3 9w 2 y 7 Rewrite the problem using a 3 6w5xy 3 9w 2 y7 division bar. Cancel the common factor, 9, in the numerator and denominator. Use the Division Property of Exponents. Simplify. 4 w5xy 3 w 7 2 y 4 w5 2 x y7 3 4 w 3 x y4 LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 391

10 Dividing a Polynomial By a Monomial When you added fractions, you learned: a c b c a b, where c 0 c If we exchange the expressions on either side of the equals sign, we have: a b a c c b c We will use this property to divide a polynomial by a monomial. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Example Find: (27w 5 x 3 y 2 12w 3 x 2 y) 3w 2 xy Solution (27w 5 x 3 y 2 12w 3 x 2 y) 3w 2 xy 27w Rewrite the problem using a 5 x 3 y 2 12w 3 x 2 y 3w division bar. 2 xy Divide each term of the 27 w5x3y 2 1 2w3x 2 y 3w2xy 3w2xy polynomial by the monomial. Cancel the common factor, 3, 9w 5x3y 2 4 w3x 2 y w2xy w2xy in each fraction. Use the Division Property of Exponents. Note that y 1 1 y w5 2 x3 1 y 2 1 4w3 2 x2 1 y w 3 x 2 y 4wx Example A landscape architect is designing a patio. She wants to estimate the cost of the patio for various widths and lengths. a. Construct an expression for the area of the patio in terms of x and y. b. If the brick she will use costs \$4.50 per square foot, find the cost of the brick for a patio that is 10 feet wide by 40 feet long. y y x 2y x 2y y y width = y length = x 392 TOPIC 6 EXPONENTS AND POLYNOMIALS

11 Solution a. The patio is made up of two triangles and a rectangle. Recall two formulas from geometry: Area of a rectangle length width Area of a triangle 1 2 (base)(height) Area lw Area 1 2 bh Express the area of each triangle in terms of y. Each triangle has base y and height y. Area 1 2 bh 1 2 (y)(y) Express the area of the rectangle in terms of x and y. 1 2 y2 Area lw The rectangle has length (x 2y) (x 2y)y and width y. xy 2y 2 The area of the patio is the area of area of area of sum of the areas of the two Area triangle rectangle triangle triangles and the rectangle. Substitute the expressions for area. 1 2 y2 xy 2y y2 Simplify. xy y 2 Therefore, the area of the patio in terms of x and y is xy y 2. b. The length of the base of the patio is x. This is 40 feet. The width of the patio, y, is 10 feet. 20 feet 10 feet 10 feet 10 feet 20 feet 10 feet length = 40 feet width = 10 feet In the formula for the area of Area xy y 2 the patio, substitute 40 feet (40 feet)(10 feet) (10 feet) 2 for x and 10 feet for y. 300 feet 2 \$ 4.50 The cost of the patio is the Cost 300 feet foot 2 4 price per square foot times the number of square feet. \$1350 The bricks for the patio will cost \$1350. LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 393

12 394 TOPIC 6 EXPONENTS AND POLYNOMIALS Here is a summary of this concept from Interactive Mathematics.

14 ❻ ❼ ❽ ❾ Multiply Monomials Find the product of monomials. Example Find: (5x 3 y)(3x 5 )(2xy 5 ) See also: Example 6.2.8, Apply Multiply a Monomial by a Polynomial Find the product of a monomial and a polynomial. Example Find: 5x 4 (3x 2 y 2 2xy 2 x 3 y) See also: Example Apply Divide a Monomial by a Monomial Find the quotient of two monomials. Example Find: 36w 5 xy 3 9w 2 y 7 See also: Apply Divide a Polynomial by a Monomial Find the quotient of a polynomial divided by a Example monomial. Find: (27w 5 x 3 y 2 12w 3 x 2 y) 3w 2 xy See also: Example Apply TOPIC 6 EXPONENTS AND POLYNOMIALS

15 Homework Homework Problems Circle the homework problems assigned to you by the computer, then complete them below. Explain Adding and Subtracting Polynomials 1. Circle the algebraic expression that is a polynomial y3 3y y3 3y y 3 3y Write m beside the monomial, b beside the binomial, and t beside the trinomial. 34x x 2 z wxy 3 z 2 pn 2 13n 3 3. Given the polynomial 3y 2y 3 4y 5 2: a. write the terms in descending order. b. find the degree of each term. c. find the degree of the polynomial. 4. Find: (3w 12w 3 2) (15w 2w 3 4w 5 3) 5. Find: (2v 3 6v 2 2) (5v v 3 4v 7 3) 6. Evaluate 1 4 xy 3y2 5x 3 when x 2 and y Find: (s 2 t s 3 t 3 4st 2 27) (3st 2 2st 8s 3 t 3 13t 36) 8. Find: (12x 3 y 9x 2 y 2 6xy y 7) (7xy x y 11x 3 y 3x 2 y 2 4) 9. Angelina works at a pet store. Today, she is cleaning three fish tanks. These polynomials describe the volumes of the tanks: Tank 1: xy 2 Tank 2: x 2 y 2y 3 4xy 2 3 Tank 3: x 2 y 5xy 2 6y 3 Write a polynomial that describes the total volume of the three tanks. Hint: Add the polynomials. volume 10. Angelina has three fish tanks to clean. These polynomials describe their volumes. Tank 1: xy 2 Tank 2: x 2 y 2y 3 4xy 2 3 Tank 3: x 2 y 5xy 2 6y 3 What is the total volume of the fish tanks if x 3 feet and y 1.5 feet? volume cubic feet 11. Find: (w 2 yz 3w 3 2wyz 2 4wyz) (4wy 2 z 3w 2 yz 2wyz 2 ) (2wyz 3) 12. Find: (tu 2 v 4t 2 u 2 v 9t 3 uv 3tv) (3t 2 u 2 2tv t 3 ) (4t 2 u 2 v 3tv 2tu 2 v) (6t 3 uv 2tv) Multiplying and Dividing Polynomials 13. Find: xyz x 2 y 2 z Find: 3p 2 r 2p 3 qr 15. Find: 6t 3 u 2 v tu2 v Find: 3y(2x 3 3x 2 y) 17. Find: 5p 2 r 3 (2pr p 2 r 2 ) 18. Find: t 3 uv 4 (2tu 3uv 4tv 5) 19. Write 12w 7 x 3 y 2 z 6 4w 2 x 2 y 3 z 6 as a fraction and simplify. LESSON 6.2 POLYNOMIAL OPERATIONS I HOMEWORK 397

17 Apply Practice Problems Here are some additional practice problems for you to try. Adding and Subtracting Polynomials 1. Circle the algebraic expressions below that are polynomials. 2xy 5xz 2 6x 3 x 9y 2 13yz 8z 2 24x 5 1 5a3 5a8 2. Circle the algebraic expressions below that are polynomials. 8xy 3 y 17x 3 3w 7wz 1 7x 2 13x 8y 2 1 2x2 3x3 3. Identify each polynomial below as a monomial, a binomial, or a trinomial. a. 17x 24z b. 13ab 2 5 c. m n 10 d. 42a 2 b 4 c e x 21y 4. Identify each polynomial below as a monomial, a binomial, or a trinomial. a. 25 6xyz 4x b. 2xyz 3 c. x y 1 d. 36 3xyz e. 32x 2 y 5. Find the degree of the polynomial 8a 3 b 5 11a 2 b 3 7b Find the degree of the polynomial 12m 4 n 7 16m Find the degree of the polynomial 7x 3 y 2 z 3x 2 y 3 z 4 6z Evaluate 2x 2 8x 11 when x Evaluate x 3 3x 2 x 1 when x Evaluate 2x 2 5x 8 when x Evaluate x 2 y xy 2 when x 2 and y Evaluate 5mn 4mn 2 8m n when m 4 and n Evaluate 3uv 6u 2 v 2u v 4 when u 2 and v Find: (3x 2 7x) (x 2 5) 15. Find: (5x 2 4x 8) (x 2 7x) 16. Find: (6a 2 8a 10) (3a 2 2a 7) 17. Find: (12m 2 n 3 7m 2 n 2 14mn) (3m 2 n 3 5m 2 n 2 7mn) 18. Find: (10x 4 y 3 9x 2 y 3 6xy 2 x) (28x 4 y 3 14x 2 y 3 3xy 2 x) 19. Find: (13a 3 b 2 6a 2 b 5ab 3 b) (2a 3 b 2 2a 2 b 4ab 3 b) 20. Find: (11u 5 v 4 w 3 6u 3 v 2 w) (6u 5 v 4 w 3 11u 3 v 2 w) 21. Find: (7xy 2 z 3 19x 2 yz 2 26x 3 y 3 z) (13xy 2 z 3 11x 2 yz 2 16x 3 y 3 z) 22. Find: (9a 4 b 2 c 3a 2 b 3 c 5abc) (2abc 6a 4 b 2 c 2) (3a 2 b 3 c 5) 23. Find: (5x 3 7x) (x 3 8) 24. Find: (9a 2 7ab 14b) (3a 2 7b) LESSON 6.2 POLYNOMIAL OPERATIONS I APPLY 399

18 25. Find: (2y 2 6xy 3y) (y 2 y) 26. Find: (8x 3 9x 2 17) (5x 3 3x 2 15) 27. Find: (9a 5 b 3 8a 4 b 6b) (2a 5 b 3 12a 4 b 3b) 28. Find: (7x 4 y 2 3x 2 y 5x) (9x 4 y 2 3x 2 y 2x) Multiplying and Dividing Polynomials 29. Find: 3y 4 5y 30. Find: 5x 3 2x 31. Find: 5a 5 9a Find: 3x 3 12x Find: 4x 3 y 5 7xy Find: 7a 5 b 6 c 3 8ab 3 c 35. Find: 3w 2 x 3 y 2 z 2x 2 yz Find: 4y 3 (3y 2 5y 10) 37. Find: 2a 3 b 2 (3a 4 b 5 5ab 3 6a) 38. Find: 2xy 3 (2x 6 5x 4 y 2 ) 39. Find: 5a 2 b 2 (4a 2 2a 2 b 7ab 2 3b) 40. Find: 4mn 3 (3m 2 n 12mn 2 6m 7n 2 ) 41. Find: 4x 3 y 3 (3x 3 7xy 2 2xy y) 42. Find: 9 x3y 3x2 43. Find: 2 0a5b 6 4a3b 44. Find: 1 2x4y 6 3x2y 32a7b9c 45. Find: 1 2a5b6c Find: 1 5m6n1 0 10n4p3 47. Find: 2 4x6y2z7 16wx3z2 48. Find: 27 a4b3c1 2 d 15ac7d3 49. Find: 4 2mn6p 3 q4 28m2nq5 50. Find: 3 6w2x3y7z 21w5y2z2 51. Find: 32a3 24a 5 8a Find: 21m2 18mn 3 3mn 53. Find: 14x 8x 4 y 2 2xy 54. Find: 24a2 b 2 c3 4ab 4 c 5 16abc Find: 32x2 y3z4 8x 5 yz 7 16x3y 3 z4 56. Find: 32r4 st2 3r 2 st 5 12r 3 s 2 t 400 TOPIC 6 EXPONENTS AND POLYNOMIALS

19 Evaluate Practice Test Take this practice test to be sure that you are prepared for the final quiz in Evaluate. 1. Circle the expressions that are polynomials p3 r 3p 2 q 2r t 2 s c c11 3 m 5 n 4 o 3 p 2 r x 2 2 3xy y 2 3 x 2. Write m beside the monomial(s), b beside the binomial(s), and t beside the trinomial(s). a. w 5 x 4 b. 2x 2 36 c. 1 3 x x d. 27 e. 27x 3 2x 2 y 3 3. Given the polynomial 3w 3 13w 2 7w 5 8w 8 2, write the terms in descending order by degree. 4. Find: a. (5x 3 y 8x 2 y 2 3xy y 3 13) (2xy 6 y 2 4y 3 2x 3 y) b. (5x 3 y 8x 2 y 2 3xy y 3 13) (2xy 6 y 2 4y 3 2x 3 y) 5. Find: x 3 y 2 w x 5 yw 4 6. Find: n 2 p 3 (3n 2n 3 p 2 35p 4 ) 7. Find: 21x 5 y 2 z 7 14xyz 8. Find: (15t 3 u 2 v 5t 5 uv 2 ) 10tuv 2 f. x 2 3xy 2 3 y2 LESSON 6.2 POLYNOMIAL OPERATIONS I EVALUATE 401

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