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Elementary Algebra 2e

6.1 Add and Subtract Polynomials

Elementary Algebra 2e6.1 Add and Subtract Polynomials
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index
Be Prepared 6.1

Before you get started, take this readiness quiz.

Simplify: 8x+3x.8x+3x.
If you missed this problem, review Example 1.24.

Be Prepared 6.2

Subtract: (5n+8)(2n1).(5n+8)(2n1).
If you missed this problem, review Example 1.139.

Be Prepared 6.3

Write in expanded form: a5.a5.
If you missed this problem, review Example 1.14.

Identify Polynomials, Monomials, Binomials and Trinomials

You have learned that a term is a constant or the product of a constant and one or more variables. When it is of the form axmaxm, where aa is a constant and mm is a whole number, it is called a monomial. Some examples of monomial are 8,−2x2,4y3,and11z78,−2x2,4y3,and11z7.

Monomials

A monomial is a term of the form axmaxm, where aa is a constant and mm is a positive whole number.

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

Polynomials

polynomial—A monomial, or two or more monomials combined by addition or subtraction, is a polynomial.

  • monomial—A polynomial with exactly one term is called a monomial.
  • binomial—A polynomial with exactly two terms is called a binomial.
  • trinomial—A polynomial with exactly three terms is called a trinomial.

Here are some examples of polynomials.

Polynomialb+14y27y+24x4+x3+8x29x+1Monomial148y2−9x3y5−13Binomiala+74b5y2163x39x2Trinomialx27x+129y2+2y86m4m3+8mz4+3z21Polynomialb+14y27y+24x4+x3+8x29x+1Monomial148y2−9x3y5−13Binomiala+74b5y2163x39x2Trinomialx27x+129y2+2y86m4m3+8mz4+3z21

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

Example 6.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

  1. 4y28y64y28y6
  2. −5a4b2−5a4b2
  3. 2x55x39x2+3x+42x55x39x2+3x+4
  4. 135m3135m3
  5. qq
Try It 6.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

5b5b 8y37y2y38y37y2y3 −3x25x+9−3x25x+9 814a2814a2 −5x6−5x6

Try It 6.2

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

27z3827z38 12m35m22m12m35m22m 5656 8x47x26x58x47x26x5 n4n4

Determine the Degree of Polynomials

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0—it has no variable.

Degree of a Polynomial

The degree of a term is the sum of the exponents of its variables.

The degree of a constant is 0.

The degree of a polynomial is the highest degree of all its terms.

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

This table has 11 rows and 5 columns. The first column is a header column, and it names each row. The first row is named “Monomial,” and each cell in this row contains a different monomial. The second row is named “Degree,” and each cell in this row contains the degree of the monomial above it. The degree of 14 is 0, the degree of 8y squared is 2, the degree of negative 9x cubed y to the fifth power is 8, and the degree of negative 13a is 1. The third row is named “Binomial,” and each cell in this row contains a different binomial. The fourth row is named “Degree of each term,” and each cell contains the degrees of the two terms in the binomial above it. The fifth row is named “Degree of polynomial,” and each cell contains the degree of the binomial as a whole.” The degrees of the terms in a plus 7 are 0 and 1, and the degree of the whole binomial is 1. The degrees of the terms in 4b squared minus 5b are 2 and 1, and the degree of the whole binomial is 2. The degrees of the terms in x squared y squared minus 16 are 4 and 0, and the degree of the whole binomial is 4. The degrees of the terms in 3n cubed minus 9n squared are 3 and 2, and the degree of the whole binomial is 3. The sixth row is named “Trinomial,” and each cell in this row contains a different trinomial. The seventh row is named “Degree of each term,” and each cell contains the degrees of the three terms in the trinomial above it. The eighth row is named “Degree of polynomial,” and each cell contains the degree of the trinomial as a whole. The degrees of the terms in x squared minus 7x plus 12 are 2, 1, and 0, and the degree of the whole trinomial is 2. The degrees of the terms in 9a squared plus 6ab plus b squared are 2, 2, and 2, and the degree of the trinomial as a whole is 2. The degrees of the terms in 6m to the fourth power minus m cubed n squared plus 8mn to the fifth power are 4, 5, and 6, and the degree of the whole trinomial is 6. The degrees of the terms in z to the fourth power plus 3z squared minus 1 are 4, 2, and 0, and the degree of the whole trinomial is 4. The ninth row is named “Polynomial,” and each cell contains a different polynomial. The tenth row is named “Degree of each term,” and the eleventh row is named “Degree of polynomial.” The degrees of the terms in b plus 1 are 1 and 0, and the degree of the whole polynomial is 1. The degrees of the terms in 4y squared minus 7y plus 2 are 2, 1, and 0, and the degree of the whole polynomial is 2. The degrees of the terms in 4x to the fourth power plus x cubed plus 8x squared minus 9x plus 1 are 4, 3, 2, 1, and 0, and the degree of the whole polynomial is 4.

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

Example 6.2

Find the degree of the following polynomials.

  1. 10y10y
  2. 4x37x+54x37x+5
  3. −15−15
  4. −8b2+9b2−8b2+9b2
  5. 8xy2+2y8xy2+2y
Try It 6.3

Find the degree of the following polynomials:

−15b−15b 10z4+4z2510z4+4z25 12c5d4+9c3d9712c5d4+9c3d97 3x2y4x3x2y4x −9−9

Try It 6.4

Find the degree of the following polynomials:

5252 a4b17a4a4b17a4 5x+6y+2z5x+6y+2z 3x25x+73x25x+7 a3a3

Add and Subtract Monomials

You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.

Example 6.3

Add: 25y2+15y225y2+15y2.

Try It 6.5

Add: 12q2+9q2.12q2+9q2.

Try It 6.6

Add: −15c2+8c2.−15c2+8c2.

Example 6.4

Subtract: 16p(−7p)16p(−7p).

Try It 6.7

Subtract: 8m(−5m).8m(−5m).

Try It 6.8

Subtract: −15z3(−5z3).−15z3(−5z3).

Remember that like terms must have the same variables with the same exponents.

Example 6.5

Simplify: c2+7d26c2c2+7d26c2.

Try It 6.9

Add: 8y2+3z23y2.8y2+3z23y2.

Try It 6.10

Add: 3m2+n27m2.3m2+n27m2.

Example 6.6

Simplify: u2v+5u23v2u2v+5u23v2.

Try It 6.11

Simplify: m2n28m2+4n2.m2n28m2+4n2.

Try It 6.12

Simplify: pq26p5q2.pq26p5q2.

Add and Subtract Polynomials

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Example 6.7

Find the sum: (5y23y+15)+(3y24y11).(5y23y+15)+(3y24y11).

Try It 6.13

Find the sum: (7x24x+5)+(x27x+3).(7x24x+5)+(x27x+3).

Try It 6.14

Find the sum: (14y2+6y4)+(3y2+8y+5).(14y2+6y4)+(3y2+8y+5).

Example 6.8

Find the difference: (9w27w+5)(2w24).(9w27w+5)(2w24).

Try It 6.15

Find the difference: (8x2+3x19)(7x214).(8x2+3x19)(7x214).

Try It 6.16

Find the difference: (9b25b4)(3b25b7).(9b25b4)(3b25b7).

Example 6.9

Subtract: (c24c+7)(c24c+7) from (7c25c+3)(7c25c+3).

Try It 6.17

Subtract: (5z26z2)(5z26z2) from (7z2+6z4)(7z2+6z4).

Try It 6.18

Subtract: (x25x8)(x25x8) from (6x2+9x1)(6x2+9x1).

Example 6.10

Find the sum: (u26uv+5v2)+(3u2+2uv)(u26uv+5v2)+(3u2+2uv).

Try It 6.19

Find the sum: (3x24xy+5y2)+(2x2xy)(3x24xy+5y2)+(2x2xy).

Try It 6.20

Find the sum: (2x23xy2y2)+(5x23xy)(2x23xy2y2)+(5x23xy).

Example 6.11

Find the difference: (p2+q2)(p2+10pq2q2)(p2+q2)(p2+10pq2q2).

Try It 6.21

Find the difference: (a2+b2)(a2+5ab6b2)(a2+b2)(a2+5ab6b2).

Try It 6.22

Find the difference: (m2+n2)(m27mn3n2)(m2+n2)(m27mn3n2).

Example 6.12

Simplify: (a3a2b)(ab2+b3)+(a2b+ab2)(a3a2b)(ab2+b3)+(a2b+ab2).

Try It 6.23

Simplify: (x3x2y)(xy2+y3)+(x2y+xy2)(x3x2y)(xy2+y3)+(x2y+xy2).

Try It 6.24

Simplify: (p3p2q)+(pq2+q3)(p2q+pq2)(p3p2q)+(pq2+q3)(p2q+pq2).

Evaluate a Polynomial for a Given Value

We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.

Example 6.13

Evaluate 5x28x+45x28x+4 when

  1. x=4x=4
  2. x=−2x=−2
  3. x=0x=0
Try It 6.25

Evaluate: 3x2+2x153x2+2x15 when

  1. x=3x=3
  2. x=−5x=−5
  3. x=0x=0
Try It 6.26

Evaluate: 5z2z45z2z4 when

  1. z=−2z=−2
  2. z=0z=0
  3. z=2z=2

Example 6.14

The polynomial −16t2+250−16t2+250 gives the height of a ball tt seconds after it is dropped from a 250 foot tall building. Find the height after t=2t=2 seconds.

Try It 6.27

The polynomial −16t2+250−16t2+250 gives the height of a ball tt seconds after it is dropped from a 250-foot tall building. Find the height after t=0t=0 seconds.

Try It 6.28

The polynomial −16t2+250−16t2+250 gives the height of a ball tt seconds after it is dropped from a 250-foot tall building. Find the height after t=3t=3 seconds.

Example 6.15

The polynomial 6x2+15xy6x2+15xy gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with x=4x=4 feet and y=6y=6 feet.

Try It 6.29

The polynomial 6x2+15xy6x2+15xy gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with x=6x=6 feet and y=4y=4 feet.

Try It 6.30

The polynomial 6x2+15xy6x2+15xy gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with x=5x=5 feet and y=8y=8 feet.

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with adding and subtracting polynomials.

Section 6.1 Exercises

Practice Makes Perfect

Identify Polynomials, Monomials, Binomials, and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

1.

81b524b3+181b524b3+1 5c3+11c2c85c3+11c2c8 1415y+171415y+17
5
4y+174y+17

2.

x2y2x2y2 −13c4−13c4 x2+5x7x2+5x7 x2y22xy+8x2y22xy+8 19

3.

83x83x z25z6z25z6 y38y2+2y16y38y2+2y16 81b524b3+181b524b3+1 −18−18

4.

11y211y2 −73−73 6x23xy+4x2y+y26x23xy+4x2y+y2 4y+174y+17 5c3+11c2c85c3+11c2c8

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

5.

6a2+12a+146a2+12a+14 18xy2z18xy2z 5x+25x+2 y38y2+2y16y38y2+2y16 −24−24

6.

9y310y2+2y69y310y2+2y6 −12p4−12p4 a2+9a+18a2+9a+18 20x2y210a2b2+3020x2y210a2b2+30 17

7.

1429x1429x z25z6z25z6 y38y2+2y16y38y2+2y16 23ab21423ab214 −3−3

8.

62y262y2 15 6x23xy+4x2y+y26x23xy+4x2y+y2 109x109x m4+4m3+6m2+4m+1m4+4m3+6m2+4m+1

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

9.

7x2+5x27x2+5x2

10.

4y3+6y34y3+6y3

11.

−12w+18w−12w+18w

12.

−3m+9m−3m+9m

13.

4a9a4a9a

14.

y5yy5y

15.

28x(−12x)28x(−12x)

16.

13z(−4z)13z(−4z)

17.

−5b17b−5b17b

18.

−10x35x−10x35x

19.

12a+5b22a12a+5b22a

20.

14x3y13x14x3y13x

21.

2a2+b26a22a2+b26a2

22.

5u2+4v26u25u2+4v26u2

23.

xy25x5y2xy25x5y2

24.

pq24p3q2pq24p3q2

25.

a2b4a5ab2a2b4a5ab2

26.

x2y3x+7xy2x2y3x+7xy2

27.

12a+8b12a+8b

28.

19y+5z19y+5z

29.

Add: 4a,−3b,−8a4a,−3b,−8a

30.

Add: 4x,3y,−3x4x,3y,−3x

31.

Subtract 5x6from12x65x6from12x6.

32.

Subtract 2p4from7p42p4from7p4.

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

33.

(5y2+12y+4)+(6y28y+7)(5y2+12y+4)+(6y28y+7)

34.

(4y2+10y+3)+(8y26y+5)(4y2+10y+3)+(8y26y+5)

35.

(x2+6x+8)+(−4x2+11x9)(x2+6x+8)+(−4x2+11x9)

36.

(y2+9y+4)+(−2y25y1)(y2+9y+4)+(−2y25y1)

37.

(8x25x+2)+(3x2+3)(8x25x+2)+(3x2+3)

38.

(7x29x+2)+(6x24)(7x29x+2)+(6x24)

39.

(5a2+8)+(a24a9)(5a2+8)+(a24a9)

40.

(p26p18)+(2p2+11)(p26p18)+(2p2+11)

41.

(4m26m3)(2m2+m7)(4m26m3)(2m2+m7)

42.

(3b24b+1)(5b2b2)(3b24b+1)(5b2b2)

43.

(a2+8a+5)(a23a+2)(a2+8a+5)(a23a+2)

44.

(b27b+5)(b22b+9)(b27b+5)(b22b+9)

45.

(12s215s)(s9)(12s215s)(s9)

46.

(10r220r)(r8)(10r220r)(r8)

47.

Subtract (9x2+2)(9x2+2) from (12x2x+6)(12x2x+6).

48.

Subtract (5y2y+12)(5y2y+12) from (10y28y20)(10y28y20).

49.

Subtract (7w24w+2)(7w24w+2) from (8w2w+6)(8w2w+6).

50.

Subtract (5x2x+12)(5x2x+12) from (9x26x20)(9x26x20).

51.

Find the sum of (2p38)(2p38) and (p2+9p+18)(p2+9p+18).

52.

Find the sum of
(q2+4q+13)(q2+4q+13) and (7q33)(7q33).

53.

Find the sum of (8a38a)(8a38a) and (a2+6a+12)(a2+6a+12).

54.

Find the sum of
(b2+5b+13)(b2+5b+13) and (4b36)(4b36).

55.

Find the difference of
(w2+w42)(w2+w42) and
(w210w+24)(w210w+24).

56.

Find the difference of
(z23z18)(z23z18) and
(z2+5z20)(z2+5z20).

57.

Find the difference of
(c2+4c33)(c2+4c33) and
(c28c+12)(c28c+12).

58.

Find the difference of
(t25t15)(t25t15) and
(t2+4t17)(t2+4t17).

59.

(7x22xy+6y2)+(3x25xy)(7x22xy+6y2)+(3x25xy)

60.

(−5x24xy3y2)+(2x27xy)(−5x24xy3y2)+(2x27xy)

61.

(7m2+mn8n2)+(3m2+2mn)(7m2+mn8n2)+(3m2+2mn)

62.

(2r23rs2s2)+(5r23rs)(2r23rs2s2)+(5r23rs)

63.

(a2b2)(a2+3ab4b2)(a2b2)(a2+3ab4b2)

64.

(m2+2n2)(m28mnn2)(m2+2n2)(m28mnn2)

65.

(u2v2)(u24uv3v2)(u2v2)(u24uv3v2)

66.

(j2k2)(j28jk5k2)(j2k2)(j28jk5k2)

67.

(p33p2q)+(2pq2+4q3)(3p2q+pq2)(p33p2q)+(2pq2+4q3)(3p2q+pq2)

68.

(a32a2b)+(ab2+b3)(3a2b+4ab2)(a32a2b)+(ab2+b3)(3a2b+4ab2)

69.

(x3x2y)(4xy2y3)+(3x2yxy2)(x3x2y)(4xy2y3)+(3x2yxy2)

70.

(x32x2y)(xy23y3)(x2y4xy2)(x32x2y)(xy23y3)(x2y4xy2)

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

71.

Evaluate 8y23y+28y23y+2 when:

y=5y=5 y=−2y=−2 y=0y=0

72.

Evaluate 5y2y75y2y7 when:

y=−4y=−4 y=1y=1 y=0y=0

73.

Evaluate 436x436x when:

x=3x=3 x=0x=0 x=−1x=−1

74.

Evaluate 1636x21636x2 when:

x=−1x=−1 x=0x=0 x=2x=2

75.

A painter drops a brush from a platform 75 feet high. The polynomial −16t2+75−16t2+75 gives the height of the brush tt seconds after it was dropped. Find the height after t=2t=2 seconds.

76.

A girl drops a ball off a cliff into the ocean. The polynomial −16t2+250−16t2+250 gives the height of a ball tt seconds after it is dropped from a 250-foot tall cliff. Find the height after t=2t=2 seconds.

77.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial −4p2+420p.−4p2+420p. Find the revenue received when p=60p=60 dollars.

78.

A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of p dollars each is given by the polynomial −4p2+420p.−4p2+420p. Find the revenue received when p=90p=90 dollars.

Everyday Math

79.

Fuel Efficiency The fuel efficiency (in miles per gallon) of a car going at a speed of xx miles per hour is given by the polynomial 1150x2+13x1150x2+13x. Find the fuel efficiency when x=30mphx=30mph.

80.

Stopping Distance The number of feet it takes for a car traveling at xx miles per hour to stop on dry, level concrete is given by the polynomial 0.06x2+1.1x0.06x2+1.1x. Find the stopping distance when x=40mphx=40mph.

81.

Rental Cost The cost to rent a rug cleaner for dd days is given by the polynomial 5.50d+255.50d+25. Find the cost to rent the cleaner for 6 days.

82.

Height of Projectile The height (in feet) of an object projected upward is given by the polynomial −16t2+60t+90−16t2+60t+90 where tt represents time in seconds. Find the height after t=2.5t=2.5 seconds.

83.

Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial 95c+3295c+32 where cc represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when c=65°.c=65°.

Writing Exercises

84.

Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

85.

Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.

86.

Ariana thinks the sum 6y2+5y46y2+5y4 is 11y611y6. What is wrong with her reasoning?

87.

Jonathan thinks that 1313 and 1x1x are both monomials. What is wrong with his reasoning?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “identify polynomials, monomials, binomials, and trinomials,” “determine the degree of polynomials,” “add and subtract monomials,” “add and subtract polynomials,” and “evaluate a polynomial for a given value.” The rest of the cells are blank.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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