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Intermediate Algebra 2e

5.1 Add and Subtract Polynomials

Intermediate Algebra 2e5.1 Add and Subtract Polynomials
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 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
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Determine the degree of polynomials
  • Add and subtract polynomials
  • Evaluate a polynomial function for a given value
  • Add and subtract polynomial functions
Be Prepared 5.1

Before you get started, take this readiness quiz.

Simplify: 3x2+3x+1+8x2+5x+5.3x2+3x+1+8x2+5x+5.
If you missed this problem, review Example 1.7.

Be Prepared 5.2

Subtract: (5n+8)(2n1).(5n+8)(2n1).
If you missed this problem, review Example 1.5.

Be Prepared 5.3

Evaluate: 4xy24xy2 when x=−2x=−2 and y=5.y=5.
If you missed this problem, review Example 1.21.

Determine the Degree of Polynomials

We have learned that a term is a constant or the product of a constant and one or more variables. A monomial is an algebraic expression with one term. When it is of the form axm,axm, where a is a constant and m is a whole number, it is called a monomial in one variable. Some examples of monomial in one variable are. Monomials can also have more than one variable such as and −4a2b3c2.−4a2b3c2.

Monomial

A monomial is an algebraic expression with one term.

A monomial in one variable is a term of the form axm,axm, where a is a constant and m is a 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 algebraic terms 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.

Polynomial y+1y+1 4a27ab+2b24a27ab+2b2 4x4+x3+8x29x+14x4+x3+8x29x+1
Monomial 14 8y28y2 −9x3y5−9x3y5 −13a3b2c−13a3b2c
Binomial a+7ba+7b 4x2y24x2y2 y216y216 3p3q9p2q3p3q9p2q
Trinomial x27x+12x27x+12 9m2+2mn8n29m2+2mn8n2 6k4k3+8k6k4k3+8k z4+3z21z4+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.

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.

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.

Let's start by looking at a monomial. The monomial 8ab28ab2 has two variables a and b. To find the degree we need to find the sum of the exponents. The variable a doesn't have an exponent written, but remember that means the exponent is 1. The exponent of b is 2. The sum of the exponents, 1+2,1+2, is 3 so the degree is 3.

The polynomial is 8 a b squared. The exponents of the variables are 1 and 2 so the degree of the monomial is 1 plus 2 which equals 3.

Here are some additional examples.

Monomial examples: 14 has degree 0, 8 a b squared has degree 3, negative 9 x cubed y to the fifth power has degree 8, negative 13 a has degree 1. Binomial examples: The terms in h plus 7 have degree 1 and 0 so the degree of the whole polynomial is 1. The terms in 7 b squared minus 3 b have degree 2 and 1 so the degree of the whole polynomial is 2. The terms in z squared y squared minus 25 have degree 4 and 0 so the degree of the whole polynomial is 4. The terms in 4 n cubed minus 8 n squared have degree 3 and 2 so the degree of the whole polynomial is 3. Trinomial examples: The terms in x squared minus 12 x plus 27 have degree 2, 1 and 0 so the degree of the whole polynomial is 2. The terms in 9 a squared plus 6 a b plus b squared have degree 2, 2, and 2 so the degree of the whole polynomial is 2. The terms in 6 m to the fourth power minus m cubed n squared plus 8 m n to the fifth power have degree 4, 5, and 6 so the degree of the whole polynomial is 6. The terms in z to the fourth power plus 3 z squared minus 1 have degree 4, 2, and 0 so the degree of the whole polynomial is 4. Polynomial examples: The terms in y minus 1 have degree 1 and 0 so the degree of the whole polynomial is 1. The terms in 3 y squared minus 2 y minus 5 have degree 2, 1, 0 so the degree of the whole polynomial is 2. The terms in 4 x to the fourth power plus x cubed plus eight x squared minus 9 x plus 1 have degree 4, 3, 2, 1, and 0 so the degree of the whole polynomial is 4.

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial. Get in the habit of writing the term with the highest degree first.

Example 5.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

7y25y+37y25y+3 −2a4b2−2a4b2 3x54x36x2+x83x54x36x2+x8 2y8xy32y8xy3 15

Try It 5.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

−5−5 8y37y2y38y37y2y3 −3x2y5xy+9xy3−3x2y5xy+9xy3 81m24n281m24n2 −3x6y3z−3x6y3z

Try It 5.2

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

64k3864k38 9m3+4m229m3+4m22 5656 8a47a3b6a2b24ab3+7b48a47a3b6a2b24ab3+7b4 p4q3p4q3

Add and Subtract Polynomials

We 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 coefficients.

Example 5.2

Add or subtract: 25y2+15y225y2+15y2 16pq3(−7pq3).16pq3(−7pq3).

Try It 5.3

Add or subtract: 12q2+9q212q2+9q2 8mn3(−5mn3).8mn3(−5mn3).

Try It 5.4

Add or subtract: −15c2+8c2−15c2+8c2 −15y2z3(−5y2z3).−15y2z3(−5y2z3).

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

Example 5.3

Simplify: a2+7b26a2a2+7b26a2 u2v+5u23v2.u2v+5u23v2.

Try It 5.5

Add: 8y2+3z23y28y2+3z23y2 m2n28m2+4n2.m2n28m2+4n2.

Try It 5.6

Add: 3m2+n27m23m2+n27m2 pq26p5q2.pq26p5q2.

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 5.4

Find the sum:(7y22y+9)+(4y28y7).(7y22y+9)+(4y28y7).

Try It 5.7

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

Try It 5.8

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

Be careful with the signs as you distribute while subtracting the polynomials in the next example.

Example 5.5

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

Try It 5.9

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

Try It 5.10

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

To subtract aa from b,b, we write it as ba,ba, placing the bb first.

Example 5.6

Subtract (p2+10pq2q2)(p2+10pq2q2) from (p2+q2).(p2+q2).

Try It 5.11

Subtract (a2+5ab6b2)(a2+5ab6b2) from (a2+b2).(a2+b2).

Try It 5.12

Subtract (m27mn3n2)(m27mn3n2) from (m2+n2).(m2+n2).

Example 5.7

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

Try It 5.13

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

Try It 5.14

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

When we add and subtract more than two polynomials, the process is the same.

Example 5.8

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

Try It 5.15

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

Try It 5.16

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

Evaluate a Polynomial Function for a Given Value

A polynomial function is a function defined by a polynomial. For example, f(x)=x2+5x+6f(x)=x2+5x+6 and g(x)=3x4g(x)=3x4 are polynomial functions, because x2+5x+6x2+5x+6 and 3x43x4 are polynomials.

Polynomial Function

A polynomial function is a function whose range values are defined by a polynomial.

In Graphs and Functions, where we first introduced functions, we learned that evaluating a function means to find the value of f(x)f(x) for a given value of x. To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.

Example 5.9

For the function f(x)=5x28x+4f(x)=5x28x+4 find: f(4)f(4) f(−2)f(−2) f(0).f(0).

Try It 5.17

For the function f(x)=3x2+2x15,f(x)=3x2+2x15, find f(3)f(3) f(−5)f(−5) f(0).f(0).

Try It 5.18

For the function g(x)=5x2x4,g(x)=5x2x4, find g(−2)g(−2) g(−1)g(−1) g(0).g(0).

The polynomial functions similar to the one in the next example are used in many fields to determine the height of an object at some time after it is projected into the air. The polynomial in the next function is used specifically for dropping something from 250 ft.

Example 5.10

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

Try It 5.19

The polynomial function h(t)=−16t2+150h(t)=−16t2+150 gives the height of a stone t seconds after it is dropped from a 150-foot tall cliff. Find the height after t=0t=0 seconds (the initial height of the object).

Try It 5.20

The polynomial function h(t)=−16t2+175h(t)=−16t2+175 gives the height of a ball t seconds after it is dropped from a 175-foot tall bridge. Find the height after t=3t=3 seconds.

Add and Subtract Polynomial Functions

Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.

Addition and Subtraction of Polynomial Functions

For functions f(x)f(x) and g(x),g(x),

(f+g)(x)=f(x)+g(x)(fg)(x)=f(x)g(x)(f+g)(x)=f(x)+g(x)(fg)(x)=f(x)g(x)

Example 5.11

For functions f(x)=3x25x+7f(x)=3x25x+7 and g(x)=x24x3,g(x)=x24x3, find:

(f+g)(x)(f+g)(x) (f+g)(3)(f+g)(3) (fg)(x)(fg)(x) (fg)(−2).(fg)(−2).

Try It 5.21

For functions f(x)=2x24x+3f(x)=2x24x+3 and g(x)=x22x6,g(x)=x22x6, find: (f+g)(x)(f+g)(x) (f+g)(3)(f+g)(3) (fg)(x)(fg)(x) (fg)(−2).(fg)(−2).

Try It 5.22

For functions f(x)=5x24x1f(x)=5x24x1 and g(x)=x2+3x+8,g(x)=x2+3x+8, find (f+g)(x)(f+g)(x) (f+g)(3)(f+g)(3) (fg)(x)(fg)(x) (fg)(−2).(fg)(−2).

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with adding and subtracting polynomials.

Section 5.1 Exercises

Practice Makes Perfect

Determine the Type of Polynomials

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

1.


47x517x2y3+y247x517x2y3+y2
5c3+11c2c85c3+11c2c8
59ab+13b59ab+13b
4
4pq+174pq+17

2.


x2y2x2y2
−13c4−13c4
a2+2ab7b2a2+2ab7b2
4x2y23xy+84x2y23xy+8
19

3.


8y5x8y5x
y25yz6z2y25yz6z2
y38y2+2y16y38y2+2y16
81ab424a2b2+3b81ab424a2b2+3b
−18−18

4.


11y211y2
−73−73
6x23xy+4x2y+y26x23xy+4x2y+y2
4y2+17z24y2+17z2
5c3+11c2c85c3+11c2c8

5.


5a2+12ab7b25a2+12ab7b2
18xy2z18xy2z
5x+25x+2
y38y2+2y16y38y2+2y16
−24−24

6.


9y310y2+2y69y310y2+2y6
−12p3q−12p3q
a2+9ab+18b2a2+9ab+18b2
20x2y210a2b2+3020x2y210a2b2+30
17

7.


14s29t14s29t
z25z6z25z6
y38y2z+2yz216z3y38y2z+2yz216z3
23ab21423ab214
−3−3

8.


15xy15xy
15
6x23xy+4x2y+y26x23xy+4x2y+y2
10p9q10p9q
m4+4m3+6m2+4m+1m4+4m3+6m2+4m+1

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

9.


7x2+5x27x2+5x2
4a9a4a9a

10.


4y3+6y34y3+6y3
y5yy5y

11.


−12w+18w−12w+18w
7x2y(−12x2y)7x2y(−12x2y)

12.


−3m+9m−3m+9m
15yz2(−8yz2)15yz2(−8yz2)

13.

7x2+5x2+4a9a7x2+5x2+4a9a

14.

4y3+6y3y5y4y3+6y3y5y

15.

−12w+18w+7x2y(−12x2y)−12w+18w+7x2y(−12x2y)

16.

−3m+9m+15yz2(−8yz2)−3m+9m+15yz2(−8yz2)

17.


−5b17b−5b17b
3xy(−8xy)+5xy3xy(−8xy)+5xy

18.


−10x35x−10x35x
17mn2(−9mn2)+3mn217mn2(−9mn2)+3mn2

19.


12a+5b22a12a+5b22a
pq24p3q2pq24p3q2

20.


14x3y13x14x3y13x
a2b4a5ab2a2b4a5ab2

21.


2a2+b26a22a2+b26a2
x2y3x+7xy2x2y3x+7xy2

22.


5u2+4v26u25u2+4v26u2
12a+8b12a+8b

23.


xy25x5y2xy25x5y2
19y+5z19y+5z

24.

12a+5b22a+pq24p3q212a+5b22a+pq24p3q2

25.

14x3y13x+a2b4a5ab214x3y13x+a2b4a5ab2

26.

2a2+b26a2+x2y3x+7xy22a2+b26a2+x2y3x+7xy2

27.

5u2+4v26u2+12a+8b5u2+4v26u2+12a+8b

28.

xy25x5y2+19y+5zxy25x5y2+19y+5z

29.

Add: 4a,−3b,−8a4a,−3b,−8a

30.

Add:4x,3y,−3x4x,3y,−3x

31.

Subtract 5x65x6 from −12x6−12x6

32.

Subtract 2p42p4 from −7p4−7p4

In the following exercises, add 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)

In the following exercises, subtract the polynomials.

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)

In the following exercises, subtract the polynomials.

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)

In the following exercises, find the difference of the polynomials.

51.

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

52.

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

In the following exercises, add the polynomials.

53.

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

54.

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

55.

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

56.

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

In the following exercises, add or subtract the polynomials.

57.

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

58.

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

59.

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

60.

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

61.

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

62.

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

Evaluate a Polynomial Function for a Given Value

In the following exercises, find the function values for each polynomial function.

63.

For the function f(x)=8x23x+2,f(x)=8x23x+2, find:
f(5)f(5) f(−2)f(−2) f(0)f(0)

64.

For the function f(x)=5x2x7,f(x)=5x2x7, find:
f(−4)f(−4) f(1)f(1) f(0)f(0)

65.

For the function g(x)=436x,g(x)=436x, find:
g(3)g(3) g(0)g(0) g(−1)g(−1)

66.

For the function g(x)=1636x2,g(x)=1636x2, find:
g(−1)g(−1) g(0)g(0) g(2)g(2)

In the following exercises, find the height for each polynomial function.

67.

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

68.

A girl drops a ball off a 200-foot cliff into the ocean. The polynomial h(t)=−16t2+200h(t)=−16t2+200 gives the height of the ball, in feet, t seconds after it is dropped. Find the height after t=3t=3 seconds.

69.

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 function R(p)=−4p2+420p.R(p)=−4p2+420p. Find the revenue received when p=60p=60 dollars.

70.

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 R(p)=−4p2+420p.R(p)=−4p2+420p. Find the revenue received when p=90p=90 dollars.

71.

The polynomial C(x)=6x2+90xC(x)=6x2+90x gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 6 feet. Find the cost of producing a box with x=4x=4 feet.

72.

The polynomial C(x)=6x2+90xC(x)=6x2+90x gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 4 feet. Find the cost of producing a box with x=6x=6 feet.

Add and Subtract Polynomial Functions

In each example, find (f + g)(x)  (f + g)(2)  (fg)(x)  (fg)(−3).

73.

f(x)=2x24x+1f(x)=2x24x+1 and g(x)=5x2+8x+3g(x)=5x2+8x+3

74.

f(x)=4x27x+3f(x)=4x27x+3 and g(x)=4x2+2x1g(x)=4x2+2x1

75.

f(x)=3x3x22x+3f(x)=3x3x22x+3 and g(x)=3x37xg(x)=3x37x

76.

f(x)=5x3x2+3x+4f(x)=5x3x2+3x+4 and g(x)=8x31g(x)=8x31

Writing Exercises

77.

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

78.

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

79.

Ariana thinks the sum 6y2+5y46y2+5y4 is 11y6.11y6. What is wrong with her reasoning?

80.

Is every trinomial a second degree polynomial? If not, give an example.

Self Check

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

The figure shows a table with six rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is "confidently", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “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”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.

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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