Download Algebraic Expressions and Equations

OpenStax-CNX module: m21848
1
Algebraic Expressions and
Equations: Classification of
Expressions and Equations
∗
Wade Ellis
Denny Burzynski
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0†
Abstract
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. Coecients are described
rather than merely dened. 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.
1 Overview
•
•
•
Polynomials
Classication of Polynomials
Classication of Polynomial Equations
2 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
3x4
Example 2
2 2 6
5x y .
A fraction occurs, but no variable appears in the denominator.
∗ Version
1.4: May 28, 2009 3:42 pm -0500
† http://creativecommons.org/licenses/by/3.0/
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OpenStax-CNX module: m21848
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Example 3
5x3 + 3x2 − 2x + 1
Some expressions that
are not
polynomials are
Example 4
3
x
− 16.
A variable appears in the denominator.
Example 5
4x2 − 5x + x−3 .
A negative exponent appears on a variable.
3 Classication of Polynomials
Polynomials can be classied using two criteria: the number of terms and degree of the polynomial.
Number of Terms
Name
Example
4x
Comment
mono
2
One
Monomial
Two
Binomial
4x2 − 7x
bi
Three
Trinomial
4x2 − 7x + 3
tri
Four or more
Polynomial
3
2
4x − 7x + 3x − 1
means one in Greek.
means two in Latin.
means three in Greek.
poly
means many in Greek.
Table 1
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 aect the degree of the term. We consider only the exponent of the variable.
For example:
Example 6
5x3
is a monomial of degree 3.
Example 7
60a5
is a monomial of degree 5.
Example 8
21b2
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
8x0 .
Since
x0 = 1 (x 6= 0), 8x0 = 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
is a monomial of the rst degree.
4xcould
be written as
4x1 .
The exponent on the variable is
sum
of the exponents of the variables, as
1 so it must be of the rst degree.
Degree of a Term Containing Several Variables
The degree of a term containing more than one variable
shown below.
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is the
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Example 11
4x2 y 5
is a monomial of degree
2 + 5 = 7.
This is a 7th degree monomial.
Example 12
37ab2 c6 d3
is a monomial of degree
1 + 2 + 6 + 3 = 12.
This is a 12th degree monomial.
Example 13
5xy
is a monomial of degree
Degree of a Polynomial
The degree of a polynomial
1 + 1 = 2.
This is a 2nd degree monomial.
is the degree of the
term
of highest degree; for example:
Example 14
2x3 + 6x − 1
is a trinomial of degree 3.
The rst term,
2x3 ,
is the term of the highest degree.
Therefore, its degree is the degree of the polynomial.
Example 15
7y − 10y 4
is a binomial of degree 4.
Example 16
a − 4 + 5a2
is a trinomial of degree 2.
Example 17
2x6 + 9x4 − x7 − 8x3 + x − 9
is a polynomial of degree 7.
Example 18
4x3 y 5 − 2xy 3
is a binomial of degree 8. The degree of the rst term is 8.
Example 19
3x + 10
is a binomial of degree 1.
Linear Quadratic Cubic
linear polynomials.
quadratic polynomials.
Polynomials of the third degree are called cubic polynomials.
Polynomials of the fourth degree are called fourth degree polynomials.
Polynomials of the nth degree are called nth degree polynomials.
Nonzero constants are polynomials of the 0th degree.
Polynomials of the rst degree are called
Polynomials of the second degree are called
Some examples of these polynomials follow:
Example 20
4x − 9
is a linear polynomial.
Example 21
3x2 + 5x − 7
is a quadratic polynomial.
Example 22
8y − 2x3
is a cubic polynomial.
Example 23
16a2 − 32a5 − 64
is a 5th degree polynomial.
Example 24
x12 − y 12
is a 12th degree polynomial.
Example 25
7x5 y 7 z 3 − 2x4 y 7 z + x3 y 7
is a 15th degree polynomial. The rst term is of degree
Example 26
43 is a 0th degree polynomial.
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5 + 7 + 3 = 15.
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4 Classication 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.
Classication 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.
5 Sample Set A
Example 27
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
5x2 + 2x − 7 = 4
is a quadratic equation since it is of degree 2.
Example 29
9x3 − 8 = 5x2 + 1 is a
” = ”sign is of degree 3.
cubic equation since it is of degree 3. The expression on the left of the
Example 30
y 4 − x4 = 0
is a 4th degree equation.
Example 31
a5 − 3a4 = −a3 + 6a4 − 7
is a 5th degree equation.
Example 32
y = 23 x + 3
is a linear equation.
Example 33
y = 3x2 − 1
is a quadratic equation.
Example 34
x2 y 2 − 4 = 0
is a 4th degree equation. The degree of
x2 y 2 − 4is 2 + 2 = 4.
6 Practice Set A
Classify the following equations in terms of their degree.
Exercise 1
(Solution on p. 8.)
3x + 6 = 0
Exercise 2
(Solution on p. 8.)
9x2 + 5x − 6 = 3
Exercise 3
(Solution on p. 8.)
25y 3 + y = 9y 2 − 17y + 4
Exercise 4
(Solution on p. 8.)
x=9
Exercise 5
(Solution on p. 8.)
y = 2x + 1
Exercise 6
3y = 9x2
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(Solution on p. 8.)
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Exercise 7
5
(Solution on p. 8.)
x2 − 9 = 0
Exercise 8
(Solution on p. 8.)
y=x
Exercise 9
(Solution on p. 8.)
5x7 = 3x5 − 2x8 + 11x − 9
7 Exercises
For the following problems, classify each polynomial as a monomial, binomial, or trinomial. State the degree
of each polynomial and write the numerical coecient of each term.
Exercise 10
(Solution on p. 8.)
5x + 7
Exercise 11
16x + 21
Exercise 12
(Solution on p. 8.)
4x2 + 9
Exercise 13
7y 3 + 8
Exercise 14
(Solution on p. 8.)
a4 + 1
Exercise 15
2b5 − 8
Exercise 16
(Solution on p. 8.)
5x
Exercise 17
7a
Exercise 18
(Solution on p. 8.)
5x3 + 2x + 3
Exercise 19
17y 4 + y 5 − 9
Exercise 20
(Solution on p. 8.)
41a3 + 22a2 + a
Exercise 21
6y 2 + 9
Exercise 22
(Solution on p. 8.)
2c6 + 0
Exercise 23
8x2 − 0
Exercise 24
(Solution on p. 8.)
9g
Exercise 25
5xy + 3x
Exercise 26
3yz − 6y + 11
http://cnx.org/content/m21848/1.4/
(Solution on p. 8.)
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Exercise 27
7ab2 c2 + 2a2 b3 c5 + a14
Exercise 28
(Solution on p. 8.)
x4 y 3 z 2 + 9z
Exercise 29
5a3 b
Exercise 30
(Solution on p. 8.)
6 + 3x2 y 5 b
Exercise 31
−9 + 3x2 + 2xy6z 2
Exercise 32
(Solution on p. 8.)
5
Exercise 33
3x2 y 0 z 4 + 12z 3 , y 6= 0
Exercise 34
(Solution on p. 8.)
4xy 3 z 5 w0 , w 6= 0
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
Exercise 36
(Solution on p. 8.)
3y − 15 = 9
Exercise 37
y = 5s + 6
Exercise 38
(Solution on p. 9.)
y = x2 + 2
Exercise 39
4y = 8x + 24
Exercise 40
(Solution on p. 9.)
9z = 12x − 18
Exercise 41
y 2 + 3 = 2y − 6
Exercise 42
(Solution on p. 9.)
y − 5 + y 3 = 3y 2 + 2
Exercise 43
x2 + x − 4 = 7x2 − 2x + 9
Exercise 44
(Solution on p. 9.)
2y + 5x − 3 + 4xy = 5xy + 2y
Exercise 45
3x − 7y = 9
Exercise 46
8a + 2b = 4b − 8
Exercise 47
2x5 − 8x2 + 9x + 4 = 12x4 + 3x3 + 4x2 + 1
http://cnx.org/content/m21848/1.4/
(Solution on p. 9.)
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Exercise 48
(Solution on p. 9.)
x−y =0
Exercise 49
x2 − 25 = 0
Exercise 50
(Solution on p. 9.)
x3 − 64 = 0
Exercise 51
x12 − y 12 = 0
Exercise 52
(Solution on p. 9.)
x + 3x5 = x + 2x5
Exercise 53
3x2 y 4 + 2x − 8y = 14
Exercise 54
(Solution on p. 9.)
10a2 b3 c6 d0 e4 + 27a3 b2 b4 b3 b2 c5 = 1, d 6= 0
Exercise 55
The expression
Exercise 56
4x3
9x−7 is not a polynomial because
(Solution on p. 9.)
The expression
a4
7−a is not a polynomial because
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
(Solution on p. 9.)
Is every polynomial expression an algebraic expression? If not, give an example of a polynomial
expression that is not an algebraic expression.
Exercise 59
How do we nd the degree of a term that contains more than one variable?
8 Exercises for Review
Exercise 60
( here1 ) Use
(Solution on p. 9.)
algebraic notation to write eleven minus three times a number is ve.
Exercise 61
5
( here2 ) Simplify x4 y 2 z 3 .
Exercise 62
( here3 ) Find
the value of
Exercise 63
( here4 ) List,
if any should appear, the common factors in the expression
Exercise 64
( here5 ) State
(Solution on p. 9.)
z if z =
x−u
s and
x = 55, u = 49,and s = 3.
3x4 + 6x3 − 18x2 .
(Solution on p. 9.)
(by writing it) the relationship being expressed by the equation
y = 3x + 5.
1 "Basic Properties of Real Numbers: Symbols and Notations" <http://cnx.org/content/m18872/latest/>
2 "Basic Properties of Real Numbers: The Power Rules for Exponents" <http://cnx.org/content/m21897/latest/>
3 "Basic Operations with Real Numbers: Multiplication and Division of Signed Numbers"
<http://cnx.org/content/m21872/latest/>
4 "Algebraic Expressions and Equations: Algebraic Expressions" <http://cnx.org/content/m18875/latest/>
5 "Algebraic Expressions and Equations: Equations" <http://cnx.org/content/m21850/latest/>
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OpenStax-CNX module: m21848
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Solutions to Exercises in this Module
Solution to Exercise (p. 4)
rst, or linear
Solution to Exercise (p. 4)
quadratic
Solution to Exercise (p. 4)
cubic
Solution to Exercise (p. 4)
linear
Solution to Exercise (p. 4)
linear
Solution to Exercise (p. 4)
quadratic
Solution to Exercise (p. 5)
quadratic
Solution to Exercise (p. 5)
linear
Solution to Exercise (p. 5)
eighth degree
Solution to Exercise (p. 5)
binomial;
rst (linear);
5, 7
Solution to Exercise (p. 5)
binomial;
second (quadratic);
4, 9
Solution to Exercise (p. 5)
binomial;
fourth;
1, 1
Solution to Exercise (p. 5)
monomial;
rst (linear);
5
Solution to Exercise (p. 5)
trinomial;
third (cubic);
5 , 2, 3
Solution to Exercise (p. 5)
trinomial;
third (cubic);
41 , 22, 1
Solution to Exercise (p. 5)
monomial;
sixth;
2
Solution to Exercise (p. 5)
monomial;
rst (linear);
9
Solution to Exercise (p. 5)
trinomial;
second (quadratic);
Solution to Exercise (p. 6)
binomial;
ninth;
1, 9
Solution to Exercise (p. 6)
binomial;
eighth;
6, 3
Solution to Exercise (p. 6)
monomial;
zero;
5
Solution to Exercise (p. 6)
monomial;
ninth;
4
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3, −6, 11
OpenStax-CNX module: m21848
Solution to Exercise (p. 6)
linear
Solution to Exercise (p. 6)
quadratic
Solution to Exercise (p. 6)
linear
Solution to Exercise (p. 6)
cubic
Solution to Exercise (p. 6)
quadratic
Solution to Exercise (p. 6)
linear
Solution to Exercise (p. 6)
linear
Solution to Exercise (p. 7)
cubic
Solution to Exercise (p. 7)
fth degree
Solution to Exercise (p. 7)
19th degree
Solution to Exercise (p. 7)
. . . there is a variable in the denominator
Solution to Exercise (p. 7)
yes
Solution to Exercise (p. 7)
11 − 3x = 5
Solution to Exercise (p. 7)
z=2
Solution to Exercise (p. 7)
The value of y is 5 more then three times the value of x.
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