LESSON 6. POLYNOMIAL OPERATIONS I LESSON 6. POLYNOMIALS OPERATIONS I 63
OVERVIEW Here's what you'll learn in this lesson: Adding and Subtracting a. Definition of polynomial, term, and coefficient b. Evaluating a polynomial c. The degree of a term and a polynomial Every day, people use algebra to find unknown quantities. For example, you may be interested in figuring out how long it would take you to drive across the country. Or, you may want to know why your checkbook doesn't balance. To find these unknown quantities, you need to be able to add, subtract, multiply, and divide polynomials. That s what you will learn in this lesson. d. Writing the terms of a polynomial in descending order e. Definition of a monomial, binomial, and trinomial f. Recognizing like or similar terms g. Combining like or similar terms h. Polynomial addition i. Polynomial subtraction Multiplying and Dividing a. Multiplying a monomial by a monomial b. Multiplying a monomial by a polynomial c. Dividing a monomial by a monomial d. Dividing a polynomial by a monomial 64 TOPIC 6 EXPONENTS AND POLYNOMIALS
EXPLAIN ADDING AND SUBTRACTING Summary Identifying Polynomials A polynomial is a special kind of algebraic expression which may have one or more variables and one or more terms. For a polynomial in one variable, x, each term has the form ax r, where the coefficient, a, is any real number, and the exponent, r, is a nonnegative integer. For example: x 3 + 7x 4x + a = 1 a = 7 a = 4 a = r = 3 r = r = 1 r = 0 Remember, when x 0: x 0 = 1 x 1 = x For a polynomial in two variables, x and y, each term has the form ax r y s, where a is any real number, and r and s are nonnegative integers. For example: 8x 3 y 4 + 3xy 4 a = 8 a = 3 a = 4 r = 3 r = 1 r = 0 s = 4 s = s = 0 Polynomials with one, two, or three terms have special names: A monomial has one term: t 5 u 3 v 1 4 A binomial has two terms: 4mpd + m 3 d A trinomial has three terms: 87k 13k + 13 k 3 An algebraic expression is not a polynomial if any of its terms cannot be written in the form ax r. For example, these algebraic expressions are not polynomials: 3w + 7w 5t r + t 3 6x y LESSON 6. POLYNOMIALS OPERATIONS I EXPLAIN 65
The Degree of a Polynomial The degree of a term of a polynomial is the sum of the exponents of the variables in that term. The degree of a polynomial is the degree of the term with the highest degree. For example, to find the degree of the polynomial 8x 3 y 4 + 3xy 3; find the degree of each term: 8x 3 y 4 + 3xy 3 = 8x 3 y 4 + 3x 1 y 3x 0 y 0 7 3 0 The degree of the polynomial 8x 3 y 4 + 3xy 3 is the degree of the term with the highest degree, 7. Evaluating Polynomials Sometimes the variables in a polynomial are assigned specific numerical values. In these cases you can evaluate the polynomial by replacing the variables with the numbers. To evaluate a polynomial: 1. Replace each variable with its assigned value.. Calculate the value of the polynomial. For example, to evaluate the polynomial 6b 4bc + c when b = and c = 3: = 6b 4bc + c 1. Replace b with and c with 3. = 6() 4()(3) + (3). Calculate. = 4 4 + 9 = 9 So, when b = and c = 3, 6b 4bc + c = 9. 66 TOPIC 6 EXPONENTS AND POLYNOMIALS
Adding Polynomials To add polynomials, combine like terms - terms that have the same variables with the same exponents. Here is an example of two like terms: Here is an example of two terms that are not like terms: 3x y and x y 3x y and 3xy To add polynomials: 1. Remove the parentheses.. Write like terms next to each other. 3. Combine like terms. For example, to find: =(3z 3 + zy 6y 3 ) + (15z 5zy + 4z ) 1. Remove the parentheses. =(3z 3 + zy 6y 3 + 15z 5zy + 4z. Write like terms next to each other. =(3z 3 + zy 5zy 6y 3 + 15z + 4z 3. Combine like terms. =(3z 3 3zy 6y 3 + 19z Subtracting Polynomials To subtract one polynomial from another, add the opposite of the polynomial being subtracted. To subtract polynomials: 1. Multiply the polynomial being subtracted by 1.. Distribute the 1. 3. Simplify. To find the opposite of a polynomial, multiply each term by 1. When you add the opposite, the result is the same as changing the sign of each term in the polynomial being subtracted. 4. Write like terms next to each other. 5. Combine like terms. For example, to find: (6y 3 3z 3 + zy ) (15z 5zy + 4y 3 ) 1. Multiply the second polynomial by 1. = (6y 3 3z 3 + zy ) + ( 1)(15z 5zy + 4y 3 ). Distribute the 1. = (6y 3 3z 3 + zy ) + ( 1)15z ( 1)5zy + ( 1)4y 3 3. Simplify. = 6y 3 3z 3 + zy 15z + 5zy 4y 3 4. Write like terms next to each other. = 6y 3 4y 3 3z 3 + zy + 5zy 15z 5. Combine like terms. = y 3 3z 3 + 7zy 15z LESSON 6. POLYNOMIALS OPERATIONS I EXPLAIN 67
Answers to Sample Problems Sample Problems 1. Evaluate the polynomial r 3 + 3s r 3s + 5 when r = 5 and s =. Evaluate: = r 3 + 3s r 3s + 5 a. Replace r with 5. = (5) 3 + 3s (5) 3s + 5 b., c. 50, 60, 6 31 b. Replace s with. = (5) 3 + 3( ) (5) 3( ) + 5 c. Calculate. = + + 5 =. Find: (x xy + ) + (3x 5xy + 3) a. Remove the parentheses. = x xy + + 3x 5xy + 3 b. 3x, 5xy, 3 c. 4x, 7xy, 5 b. Write like terms next to = x + xy + + each other. c. Combine like terms. = + 3. Find: (x xy + ) (3x 5xy + 3) a. Multiply each term in the second polynomial by 1. = (x xy + ) + ( 1)(3x 5xy + 3) b. 3x, 5xy, 3 c. x xy + 3x + 5xy 3 d. x 3x xy + 5xy + 3 (in any order that like terms are next to each other) e. x + 3xy 1 (in any order) b. Distribute the 1. = (x xy + ) + ( 1)( ) + ( 1)( ) + ( 1)( ) c. Simplify. = d. Write like terms next to each other. = e. Combine like terms. = 68 TOPIC 6 EXPONENTS AND POLYNOMIALS
MULTIPLYING AND DIVIDING Summary Multiplying Monomials Monomials are easy to multiply because they each have only one term. To multiply monomials: 1. Rearrange the factors so that the constants are next to each other and the factors with the same base are next to each other.. Multiply. For example, to find: 3p 3 q 6 r s p 4 q 4 s 5 1. Rearrange the factors. = 3 p 3 p 4 q 6 q 4 r s s 5. Multiply. = 6 p 3 + 4 q 6 + 4 r s 1 + 5 = 6 p 7 q 10 r s 6 = 6p 7 q 10 r s 6 Multiplying a Monomial by a Polynomial with More Than One Term When multiplying a monomial by a polynomial with more than one term, you need to multiply every term in the polynomial by the monomial. To multiply a monomial by a polynomial with more than one term: 1. Distribute the monomial to each term in the other polynomial.. Simplify. In general, to multiply a monomial by a polynomial with more than one term: a(b + c + d) = a b + a c + a d For example, to find: 5x 4 (3x y + xy x y ) 1. Distribute the = 5x 4 (3x y ) + 5x 4 (xy ) 5x 4 (x y ) monomial. = 5 3 x 4 x y + 5 x 4 x y 5 x 4 x y. Simplify. = 15 x 4+ y + 10 x 4 + 1 y 5 x 4 + y = 15x 6 y + 10x 5 y 5x 6 y LESSON 6. POLYNOMIALS OPERATIONS I EXPLAIN 69
Remember: x r x s When r > s, x r x s = = x r s. When r < s, x r x s = x r = 1. x s r x s Dividing Monomials To divide monomials: 1. Write the division as a fraction.. Cancel common numerical factors. 3. Divide factors with the same base by subtracting exponents. For example, to find 4x 3 y 4 z 5 10x y 6 z : 1. Write the division as a fraction. =. Cancel common numerical factors. = 4x 3 y 4 z 5 10x y 6 z 1 x 3 y 4 z 5 / x 3 y 4 z 5 / 5x y 6 z 1 x 3 y 4 z 5 The y is on the bottom since the exponent of y in the denominator is bigger than the exponent of y in the numerator. 3. Divide factors with the same base. = Dividing a Polynomial with More Than One Term by a Monomial When you divide a polynomial with more than one term by a monomial, you must divide each term of the polynomial by the monomial. a b c Use this rule: = +. a c b c = x 3 z 5 5y 6 4 x 1 z 3 5y To divide a polynomial with more than one term by a monomial: 1. Write the division as a fraction.. Rewrite the fraction using the rule = +. 3. Perform the division on each of the resulting terms. For example, to find (15x 6 y + 10x 5 y 3 ) 5x 4 y : 1. Write the division as a fraction. =. Rewrite the fraction using a b a b 15x the rule = +. = 6 y c + c c a b c a c 15x 6 y 10x 5 y 3 5x 4 y 5x 4 y b c 10x 5 y 3 5x 4 y 3 1x 6 4 y 1 3 5/x 6 4 y 1 5/ 1 3. Divide each of the resulting terms. = + 51x 5 4 y 3 1 5/ x 5 4 y 3 1 5/ 1 = 3x y + xy 70 TOPIC 6 EXPONENTS AND POLYNOMIALS
Sample Problems Answers to Sample Problems 1. Find: 3wx 3 y 6 7wx y 5 z a. Rearrange the factors so the constants are next to each other and factors with the same base are next to each other. = 3 7 w w x 3 x y 6 y 5 z b. Multiply factors with the same base by adding the exponents. = 1 w x y z b., 5, 11,. Find: pr s(p r + pr 3 s 6 ) a. Distribute the monomial. = p r + pr 3 s 6 a. pr s, pr s, pr s b. Multiply each of the resulting terms. = p 1+ r +1 s + p 1+1 r +3 s 1+6 pr s = + pr s b. p 3 r 3 s, p r 5 s 7 3. Find: 18m 6 n 5 p 3 r 10m 3 n 7 pr a. Write the division as a fraction. = 18m 6 n 5 p 3 r 10m 3 n 7 pr b. Cancel common numerical factors. = 1 / 9m 6 n 5 p 3 r / 5m 3 n 7 pr 1 c. Divide factors with the same base by = subtracting exponents. 4. Find: (6w 5 x 3 + 4w 3 x y) w xy a. Write the division as a fraction. = b. Rewrite the fraction using the rule a b = + c. a b c c 6w 5 x 3 4w 3 x y w xy 6w 5 x 3 w xy = + c. b. 9m 3 p 5n 4w 3 x y w xy c. Divide each term. 3w = 3 x + y c. wx LESSON 6. POLYNOMIALS OPERATIONS I EXPLAIN 71
HOMEWORK Homework Problems Circle the homework problems assigned to you by the computer, then complete them below. Explain Adding and Subtracting 1. Circle the algebraic expression that is a polynomial. 1 4 3 y 3 + 3y 5 1 4 1 4y 3 3 y 3 + 3y 5 + 3y 5. Write m beside the monomial, b beside the binomial, and t beside the trinomial. 34x + x + z wxy 3 z pn 13n 3 3. Given the polynomial 3y y 3 4y 5 + : a. write the terms in descending order. b. find the degree of each term. c. find the degree of the polynomial. 4. Find: ( 3w 1w 3 + ) + (15w w 3 + 4w 5 3) 5. Find: (v 3 + 6v + ) (5v + v 3 + 4v 7 3) 1 4 6. Evaluate xy + 3y 5x 3 when x = and y = 4. 7. Find: ( s t + s 3 t 3 + 4st 7) + (3st + st 8s 3 t 3 13t + 36) 8. Find: (1x 3 y + 9x y + 6xy y + 7) (7xy x + y 11x 3 y + 3x y 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 Tank : x y y 3 + 4xy + 3 Tank 3: x y + 5xy + 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 Tank : x y y 3 + 4xy + 3 Tank 3: x y + 5xy + 6y 3 What is the total volume of the fish tanks if x = 3 feet and y = 1.5 feet? volume = cubic feet 7 TOPIC 6 EXPONENTS AND POLYNOMIALS
11. Find: (w yz + 3w 3 wyz + 4wyz) (4wy z 3w yz + wyz ) + (wyz + 3) 1. Find: (tu v 4t u v + 9t 3 uv + 3tv) + (3t u + tv t 3 ) (4t u v + 3tv + tu v) (6t 3 uv + tv) Multiplying and Dividing 13. Find: xyz x y z 14. Find: 3p r p 3 qr 15. Find: 6t 3 u v 11 tu v 4 1 16. Find: 3y(x 3 + 3x y) 17. Find: 5p r 3 (pr + p r ) 18. Find: t 3 uv 4 (tu 3uv + 4tv + 5) 19. Write 1w 7 x 3 y z 6 4w x y 3 z 6 as a fraction and simplify. 0. Write (36x 3 y 3 + 15x y 5 ) 9x y as a fraction and simplify. 1. Find: 15a 7 b 4 d 10a 4 b 9 c 3 d. Tony is an algebra student. This is how he answered a question on a test: (t 8 u 3 4t 4 u 9 + 6t 1 u 6 ) t 4 u 3 = t u tu 3 + 3t 3 u Is his answer right or wrong? Why? Circle the most appropriate response. The answer is right. The answer is wrong. Tony divided the exponents rather than adding them. The correct answer is t 1 u 6 t 8 u 1 + t 16 u 9. The answer is wrong. The terms need to be ordered by degree. The correct answer is 3t 3 u + t u tu 3. The answer is wrong. Tony divided the exponents rather than subtracting them. The correct answer is t 4 u 6 + 3t 8 u 3. The answer is wrong. Tony shouldn't have canceled the numerical coefficients. The correct answer is t u 4tu 3 + 6t 3 u. 3. Find: (16x y 4 + 0x 3 y 5 ) 1xy 4. Find: (0t 5 u 11 + 5t 3 u 5 + 30tu 6 v 5 ) 10t 4 u 5 LESSON 6. POLYNOMIALS OPERATIONS I HOMEWORK 73
APPLY Practice Problems Here are some additional practice problems for you to try. Adding and Subtracting 1. Circle the algebraic expressions below that are polynomials. xy 5xz 3x + 6x 9y + 13yz 8z 4 x 5 15a 3 5a 8. Circle the algebraic expressions below that are polynomials. 8xy 17 x 3 3w 7wz 1 7x 13x 8y 1x 3x 3 3. Identify each polynomial below as a monomial, a binomial, or a trinomial. a. 17x 4z b. 13ab 5 c. m n 10 d. 4a b 4 c 3 y e. 73 65x 1y 4. Identify each polynomial below as a monomial, a binomial, or a trinomial. a. 5 6xyz 4x b. xyz 3 c. x y 1 d. 36 3xyz e. 3x y 5. Find the degree of the polynomial 8a 3 b 5 11a b 3 7b 6. 6. Find the degree of the polynomial 1m 4 n 7 16m 1. 7. Find the degree of the polynomial 7x 3 y z 3x y 3 z 4 6z 7. 8. Evaluate x 8x + 11 when x = 1. 9. Evaluate x 3 + 3x x + 1 when x =. 10. Evaluate x 5x + 8 when x = 3. 11. Evaluate x y + xy when x = and y = 3. 1. Evaluate 5mn 4mn 8m n when m 4 and n. 13. Evaluate 3uv 6u v u v 4 when u and v 4. 14. Find: (3x + 7x ) + (x 5) 15. Find: (5x + 4x 8) + (x + 7x ) 16. Find: (6a + 8a 10) ( 3a a + 7) 74 TOPIC 6 EXPONENTS AND POLYNOMIALS
17. Find: (1m n 3 7m n 14mn ) (3m n 3 5m n 7mn ) 18. Find: (10x 4 y 3 9x y 3 + 6xy x ) + ( 8x 4 y 3 + 14x y 3 + 3xy + x ) 19. Find: (13a 3 b 6a b 5ab 3 b ) (a 3 b a b 4ab 3 b ) 0. Find: (11u 5 v 4 w 3 6u 3 v w) (6u 5 v 4 w 3 11u 3 v w ) 1. Find: (7xy z 3 19x yz 6x 3 y 3 z ) (13xy z 3 11x yz 16x 3 y 3 z ). Find: (9a 4 b c 3a b 3 c 5abc) (abc 6a 4 b c ) (3a b 3 c 5) 3. Find: (5x 3 7x ) (x 3 8) 4. Find: (9a 7ab 14b ) (3a 7b ) 5. Find: (y 6xy 3y ) (y y ) 6. Find: (8x 3 9x 17) (5x 3 3x 15) 7. Find: (9a 5 b 3 8a 4 b 6b ) ( a 5 b 3 1a 4 b 3b ) 8. Find: (7x 4 y 3x y 5x ) (9x 4 y 3x y x ) Multiplying and Dividing 9. Find: 3y 4 5y 30. Find: 5x 3 x 31. Find: 5a 5 9a 4 3. Find: 3x 3 1x 4 33. Find: 4x 3 y 5 7xy 3 34. Find: 7a 5 b 6 c 3 8ab 3 c 35. Find: 3w x 3 y z x yz 36. Find: 4y 3 (3y 5y 10) 37. Find: a 3 b (3a 4 b 5 5ab 3 6a ) 38. Find: xy 3 (x 6 5x 4 y ) 39. Find: 5a b (4a a b 7ab 3b ) 40. Find: 4mn 3 ( 3m n 1mn 6m 7n ) 41. Find: 4x 3 y 3 (3x 3 7xy xy y ) 4. Find: 43. Find: 44. Find: 45. Find: 46. Find: 47. Find: 48. Find: 49. Find: 50. Find: 51. Find: 5. Find: 53. Find: 54. Find: 55. Find: 9x 3 y 3x 0a 5 b 6 4a 3 b 1x 4 y 6 3x y 3a 7 b 9 c 1a 5 b 6 c 15m 6 n 10 10n 4 p 3 4x 6 y z 7 16wx 3 z 7a 4 b 3 c 1 d 15ac 7 d 3 4mn 6 p 3 q 4 8m nq 5 36w x 3 y 7 z 1w 5 y z 3a 3 4a 5 8a 1m 18mn 3 3mn 14x 8x 4 y xy 4a b c 3 4ab 4 c 5 16abc 3 3x y 3 z 4 8x 5 yz 7 16x 3 y 3 z 4 56. Find: 3r 4 st 3r st 5 1r 3 s t LESSON 6. POLYNOMIALS OPERATIONS I APPLY 75
Practice Test EVALUATE Take this practice test to be sure that you are prepared for the final quiz in Evaluate. 1. Circle the expressions that are polynomials. 3 5 p 3 r 3p q + r t s + 5 c 15 + c 11 3π m 5 n 4 o 3 p r x + 3xy + y. Write m beside the monomial(s), b beside the binomial(s), and t beside the trinomial(s). a. w 5 x 4 b. x 36 1 3 c. x 17 + x 1 d. 7 e. 7x 3 x y 3 f. x + 3xy y 3 3 1 3 5 5 7 3 14 3x 3. Given the polynomial 3w 3 13w + 7w 5 + 8w 8, write the terms in descending order by degree. 4. Find: a. (5x 3 y 8x y + 3xy y 3 + 13) + ( xy + 6 + y 4y 3 x 3 y) b. (5x 3 y 8x y + 3xy y 3 + 13) ( xy + 6 + y 4y 3 x 3 y) 5. Find: x 3 y w x 5 yw 4 6. Find: n p 3 (3n + n 3 p 35p 4 ) 7. Find: 1x 5 y z 7 14xyz 8. Find: (15t 3 u v 5t 5 uv ) 10tuv as 76 TOPIC 6 EXPONENTS AND POLYNOMIALS