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Classification of Expressions and Equations

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form. Objectives of this module: be familar with polynomials, be able classify polynomials and polynomial equations.

Overview

  • Polynomials
  • Classification of Polynomials
  • Classification of Polynomial Equations

Polynomials

Polynomials

Let us consider the collection of all algebraic expressions that do not contain variables in the denominators of fractions and where all exponents on the variable quantities are whole numbers. Expressions in this collection are called polynomials.

Some expressions that are polynomials are

Example 1

3 x 4 3 x 4

Example 2

2 5 x 2 y 6 2 5 x 2 y 6 .

A fraction occurs, but no variable appears in the denominator.

Example 3

5 x 3 +3 x 2 −2x+1 5 x 3 +3 x 2 −2x+1

Some expressions that are not polynomials are

Example 4

3 x −16 3 x −16 .

A variable appears in the denominator.

Example 5

4 x 2 −5x+ x −3 4 x 2 −5x+ x −3 .

A negative exponent appears on a variable.

Classification of Polynomials

Polynomials can be classified using two criteria: the number of terms and degree of the polynomial.

Table 1
Number of Terms Name Example Comment
One Monomial 4 x 2 4 x 2 mono means “one” in Greek.
Two Binomial 4 x 2 −7x 4 x 2 −7x bi means “two” in Latin.
Three Trinomial 4 x 2 −7x+3 4 x 2 −7x+3 tri means “three” in Greek.
Four or more Polynomial 4 x 3 −7 x 2 +3x−1 4 x 3 −7 x 2 +3x−1 poly means “many” in Greek.

Degree of a Term Containing One Variable

The degree of a term containing only one variable is the value of the exponent of the variable. Exponents appearing on numbers do not affect the degree of the term. We consider only the exponent of the variable. For example:

Example 6

5 x 3 5 x 3 is a monomial of degree 3.

Example 7

60 a 5 60 a 5 is a monomial of degree 5.

Example 8

21 b 2 21 b 2 is a monomial of degree 2.

Example 9

8 is a monomial of degree 0. We say that a nonzero number is a term of 0 degree since it could be written as 8 x 0 8 x 0 . Since x 0 =1(x≠0) x 0 =1(x≠0) , 8 x 0 =8 8 x 0 =8 . The exponent on the variable is 0 so it must be of degree 0. (By convention, the number 0 has no degree.)

Example 10

4x 4x is a monomial of the first degree. 4x 4x could be written as 4 x 1 4 x 1 . The exponent on the variable is 1 so it must be of the first degree.

Degree of a Term Containing Several Variables

The degree of a term containing more than one variable is the sum of the exponents of the variables, as shown below.

Example 11

4 x 2 y 5 4 x 2 y 5 is a monomial of degree 2+5=7 2+5=7 . This is a 7th degree monomial.

Example 12

37a b 2 c 6 d 3 37a b 2 c 6 d 3 is a monomial of degree 1+2+6+3=12 1+2+6+3=12 . This is a 12th degree monomial.

Example 13

5xy 5xy is a monomial of degree 1+1=2 1+1=2 . This is a 2nd degree monomial.

Degree of a Polynomial

The degree of a polynomial is the degree of the term of highest degree; for example:

Example 14

2 x 3 +6x−1 2 x 3 +6x−1 is a trinomial of degree 3. The first term, 2 x 3 2 x 3 , is the term of the highest degree. Therefore, its degree is the degree of the polynomial.

Example 15

7y−10 y 4 7y−10 y 4 is a binomial of degree 4.

Example 16

a−4+5 a 2 a−4+5 a 2 is a trinomial of degree 2.

Example 17

2 x 6 +9 x 4 − x 7 −8 x 3 +x−9 2 x 6 +9 x 4 − x 7 −8 x 3 +x−9 is a polynomial of degree 7.

Example 18

4 x 3 y 5 −2x y 3 4 x 3 y 5 −2x y 3 is a binomial of degree 8. The degree of the first term is 8.

Example 19

3x+10 3x+10 is a binomial of degree 1.

Linear Quadratic Cubic

Polynomials of the first degree are called linear polynomials.
Polynomials of the second degree are called quadratic polynomials.
Polynomials of the third degree are called cubic polynomials.
Polynomials of the fourth degree are called fourth degree polynomials.
Polynomials of the n n th degree are called n n th degree polynomials.
 Nonzero constants are polynomials of the 0th degree.

Some examples of these polynomials follow:

Example 20

4x−9 4x−9 is a linear polynomial.

Example 21

3 x 2 +5x−7 3 x 2 +5x−7 is a quadratic polynomial.

Example 22

8y−2 x 3 8y−2 x 3 is a cubic polynomial.

Example 23

16 a 2 −32 a 5 −64 16 a 2 −32 a 5 −64 is a 5th degree polynomial.

Example 24

x 12 − y 12 x 12 − y 12 is a 12th degree polynomial.

Example 25

7 x 5 y 7 z 3 −2 x 4 y 7 z+ x 3 y 7 7 x 5 y 7 z 3 −2 x 4 y 7 z+ x 3 y 7 is a 15th degree polynomial. The first term is of degree 5+7+3=15 5+7+3=15 .

Example 26

43 is a 0th degree polynomial.

Classification of Polynomial Equations

As we know, an equation is composed of two algebraic expressions separated by an equal sign. If the two expressions happen to be polynomial expressions, then we can classify the equation according to its degree. Classification of equations by degree is useful since equations of the same degree have the same type of graph. (We will study graphs of equations in Chapter 6.)

The degree of an equation is the degree of the highest degree expression.

Sample Set A

Example 27

x+7=15 x+7=15 .

This is a linear equation since it is of degree 1, the degree of the expression on the left of the "=" "=" sign.

Example 28

5 x 2 +2x−7=4 5 x 2 +2x−7=4 is a quadratic equation since it is of degree 2.

Example 29

9 x 3 −8=5 x 2 +1 9 x 3 −8=5 x 2 +1 is a cubic equation since it is of degree 3. The expression on the left of the "=" "=" sign is of degree 3.

Example 30

y 4 − x 4 =0 y 4 − x 4 =0 is a 4th degree equation.

Example 31

a 5 −3 a 4 =− a 3 +6 a 4 −7 a 5 −3 a 4 =− a 3 +6 a 4 −7 is a 5th degree equation.

Example 32

y= 2 3 x+3 y= 2 3 x+3 is a linear equation.

Example 33

y=3 x 2 −1 y=3 x 2 −1 is a quadratic equation.

Example 34

x 2 y 2 −4=0 x 2 y 2 −4=0 is a 4th degree equation. The degree of x 2 y 2 −4 x 2 y 2 −4 is 2+2=4 2+2=4 .

Practice Set A

Classify the following equations in terms of their degree.

Exercise 1

3x+6=0 3x+6=0

Solution

first, or linear

Exercise 2

9 x 2 +5x−6=3 9 x 2 +5x−6=3

Solution

quadratic

Exercise 3

25 y 3 +y=9 y 2 −17y+4 25 y 3 +y=9 y 2 −17y+4

Solution

cubic

Exercise 4

x=9 x=9

Solution

linear

Exercise 5

y=2x+1 y=2x+1

Solution

linear

Exercise 6

3y=9 x 2 3y=9 x 2

Solution

quadratic

Exercise 7

x 2 −9=0 x 2 −9=0

Solution

quadratic

Exercise 8

y=x y=x

Solution

linear

Exercise 9

5 x 7 =3 x 5 −2 x 8 +11x−9 5 x 7 =3 x 5 −2 x 8 +11x−9

Solution

eighth degree

Exercises

For the following problems, classify each polynomial as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term.

Exercise 10

5x+7 5x+7

Solution

binomial; first (linear); 5,7 binomial; first (linear); 5,7

Exercise 11

16x+21 16x+21

Exercise 12

4 x 2 +9 4 x 2 +9

Solution

binomial; second (quadratic); 4,9 binomial; second (quadratic); 4,9

Exercise 13

7 y 3 +8 7 y 3 +8

Exercise 14

a 4 +1 a 4 +1

Solution

binomial; fourth; 1,1 binomial; fourth; 1,1

Exercise 15

2 b 5 −8 2 b 5 −8

Exercise 16

5x 5x

Solution

monomial; first (linear); 5 monomial; first (linear); 5

Exercise 17

7a 7a

Exercise 18

5 x 3 +2x+3 5 x 3 +2x+3

Solution

trinomial; third (cubic); 5 ,2,3 trinomial; third (cubic); 5 ,2,3

Exercise 19

17 y 4 + y 5 −9 17 y 4 + y 5 −9

Exercise 20

41 a 3 +22 a 2 +a 41 a 3 +22 a 2 +a

Solution

trinomial; third (cubic); 41 ,22,1 trinomial; third (cubic); 41 ,22,1

Exercise 21

6 y 2 +9 6 y 2 +9

Exercise 22

2 c 6 +0 2 c 6 +0

Solution

monomial; sixth; 2 monomial; sixth; 2

Exercise 23

8 x 2 −0 8 x 2 −0

Exercise 24

9g 9g

Solution

monomial; first (linear); 9 monomial; first (linear); 9

Exercise 25

5xy+3x 5xy+3x

Exercise 26

3yz−6y+11 3yz−6y+11

Solution

trinomial; second (quadratic); 3,−6,11 trinomial; second (quadratic); 3,−6,11

Exercise 27

7a b 2 c 2 +2 a 2 b 3 c 5 + a 14 7a b 2 c 2 +2 a 2 b 3 c 5 + a 14

Exercise 28

x 4 y 3 z 2 +9z x 4 y 3 z 2 +9z

Solution

binomial; ninth; 1,9 binomial; ninth; 1,9

Exercise 29

5 a 3 b 5 a 3 b

Exercise 30

6+3 x 2 y 5 b 6+3 x 2 y 5 b

Solution

binomial; eighth; 6,3 binomial; eighth; 6,3

Exercise 31

−9+3 x 2 +2xy6 z 2 −9+3 x 2 +2xy6 z 2

Exercise 32

5

Solution

monomial; zero; 5 monomial; zero; 5

Exercise 33

3 x 2 y 0 z 4 +12 z 3 ,   y≠0 3 x 2 y 0 z 4 +12 z 3 ,   y≠0

Exercise 34

4x y 3 z 5 w 0 ,   w≠0 4x y 3 z 5 w 0 ,   w≠0

Solution

monomial; ninth; 4 monomial; ninth; 4

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it.

Exercise 35

4x+7=0 4x+7=0

Exercise 36

3y−15=9 3y−15=9

Solution

linear

Exercise 37

y=5s+6 y=5s+6

Exercise 38

y= x 2 +2 y= x 2 +2

Solution

quadratic

Exercise 39

4y=8x+24 4y=8x+24

Exercise 40

9z=12x−18 9z=12x−18

Solution

linear

Exercise 41

y 2 +3=2y−6 y 2 +3=2y−6

Exercise 42

y−5+ y 3 =3 y 2 +2 y−5+ y 3 =3 y 2 +2

Solution

cubic

Exercise 43

x 2 +x−4=7 x 2 −2x+9 x 2 +x−4=7 x 2 −2x+9

Exercise 44

2y+5x−3+4xy=5xy+2y 2y+5x−3+4xy=5xy+2y

Solution

quadratic

Exercise 45

3x−7y=9 3x−7y=9

Exercise 46

8a+2b=4b−8 8a+2b=4b−8

Solution

linear

Exercise 47

2 x 5 −8 x 2 +9x+4=12 x 4 +3 x 3 +4 x 2 +1 2 x 5 −8 x 2 +9x+4=12 x 4 +3 x 3 +4 x 2 +1

Exercise 48

x−y=0 x−y=0

Solution

linear

Exercise 49

x 2 −25=0 x 2 −25=0

Exercise 50

x 3 −64=0 x 3 −64=0

Solution

cubic

Exercise 51

x 12 − y 12 =0 x 12 − y 12 =0

Exercise 52

x+3 x 5 =x+2 x 5 x+3 x 5 =x+2 x 5

Solution

fifth degree

Exercise 53

3 x 2 y 4 +2x−8y=14 3 x 2 y 4 +2x−8y=14

Exercise 54

10 a 2 b 3 c 6 d 0 e 4 +27 a 3 b 2 b 4 b 3 b 2 c 5 =1, d≠0 10 a 2 b 3 c 6 d 0 e 4 +27 a 3 b 2 b 4 b 3 b 2 c 5 =1, d≠0

Solution

19th degree

Exercise 55

The expression 4 x 3 9x−7 4 x 3 9x−7 is not a polynomial because

Exercise 56

The expression a 4 7−a a 4 7−a is not a polynomial because

Solution

. . . there is a variable in the denominator

Exercise 57

Is every algebraic expression a polynomial expression? If not, give an example of an algebraic expression that is not a polynomial expression.

Exercise 58

Is every polynomial expression an algebraic expression? If not, give an example of a polynomial expression that is not an algebraic expression.

Solution

yes

Exercise 59

How do we find the degree of a term that contains more than one variable?

Exercises for Review

Exercise 60

((Reference)) Use algebraic notation to write “eleven minus three times a number is five.”

Solution

11−3x=5 11−3x=5

Exercise 61

((Reference)) Simplify ( x 4 y 2 z 3 ) 5 ( x 4 y 2 z 3 ) 5 .

Exercise 62

((Reference)) Find the value of z z if z= x−u s z= x−u s and x=55, u=49, x=55, u=49, and s=3 s=3 .

Solution

z=2 z=2

Exercise 63

((Reference)) List, if any should appear, the common factors in the expression 3 x 4 +6 x 3 −18 x 2 3 x 4 +6 x 3 −18 x 2 .

Exercise 64

((Reference)) State (by writing it) the relationship being expressed by the equation y=3x+5 y=3x+5 .

Solution

The value​ of y is 5 more then three times the value of x. The value​ of y is 5 more then three times the value of x.

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