4.4: Classification of Expressions and 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
\(3x^4\)
\(\dfrac{2}{5}x^2y^6\)
A fraction occurs, but no variable appears in the denominator.
\(5x^3+3x^2-2x+1\)
Some expressions that are not polynomials are
\(\dfrac{3}{x}-16\)
A variable appears in the denominator
\(4x^2 - 5x + x^{-3}\)
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.
| Number of Terms | Name | Example | Comment |
| One | Monmial | \(4x^2\) | mono means "one" in Greek |
| Two | Binomial | \(4x^2 - 7x\) | bi means "two" in Latin |
| Three | Trinomial | \(4x^2-7x+3\) | Tri means "three" in Greek |
| Four or more | Polynomial | \(4x^3-7x^2+3x-1\) | 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:
\(5x^3\) is a monomial of degree \(3\).
\(60a^5\) is a monomial of degree \(5\).
\(21b^2\) is a monomial of degree \(2\)
\(8\) is a monomial of degree 0. We say that a nonzero number is a term of \(0\) degree since it could be written as \(8x^0\). Since \(x^0=1(x\not =0)\), \(8x^0=8\). The exponent on the variable is \(0\) so it must be of degree \(0\). (By convention, the number \(0\) has no degree.)
\(4x\) is a monomial of the first degree. \(4x\) could be written as \(4x^1\). 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.
\(4x^2y^5\) is a monomial of degree \(2 + 5 = 7\). This is a 7th degree monomial.
\(37ab^2c^6d^3\) is a monomial of degree \(1 + 2 + 6 + 3 = 12\). This is a 12th degree monomial.
\(5xy\) is a monomial of degree \(1 + 1 = 2\). 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:
\(2x^3 + 6x - 1\) is a trinomial of degree \(3\). The first term, \(2x^3\), is the term of the highest degree. Therefore, its degree is the degree of the polynomial.
\(7y -10y^4\) is a binomial of degree \(4\).
\(a - 4 + 5a^2\) is a trinomial of degree \(2\).
\(2x^6 + 9x^4 - x^7 - 8x^3 + x - 9\) is a polynomial of degree \(7\).
\(4x^3y^5 - 2xy^3\) is a binomial of degree \(8\). The degree of the first term is \(8\).
\(3x + 10\) 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 nth degree are called \(n\)
th degree
polynomials.
Nonzero constants
are polynomials of the
0th
degree.
Some examples of these polynomials follow:
\(4x - 9\) is a linear polynomial.
\(3x^2 + 5x - 7\) is a quadratic polynomial.
\(8y - 2x^3\) is a cubic polynomial
\(16a^2 - 32a^5 - 64\) is a 5th degree polynomial.
\(x^{12} - y^{12}\) is a 12th degree polynomial.
\(7x^5y^7z^3 - 2x^4y^7z + x^3y^7\) is a 15th degree polynomial. The first term is of degree \(5 + 7 + 3 = 15\).
\(43\) 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 8.)
The degree of an equation is the degree of the highest degree expression.
Sample Set A
\(x + 7 = 15\).
This is a linear equation since it is of degree 1, the degree of the expression on the left of the "=" sign.
\(5x^2 + 2x - 7 = 4\) is a quadratic equation since it is of degree 2.
\(9x^3 - 8 = 5x^2 + 1\)
\(y^4 - x^4 = 0\) is a 4th degree equation.
\(a^5 - 3a^4 = -a^3 + 6a^4 - 7\) is a 5th degree equation.
\(y = \dfrac{2}{3}x + 3\) is a linear equation.
\(y = 3x^2 - 1\) is a quadratic equation.
\(x^2y^2 - 4 = 0\) ios a 4th degree equation. The degree of \(x^2y^2 - 4\) is \(2 + 2 = 4\).
Practice Set A
Classify the following equations in terms of their degree.
\(3x + 6 = 0\)
- Answer
-
First, or linear
\(9x^2 + 5x - 6 = 3\)
- Answer
-
Quadratic
\(25y^3 + y = 9y^2 - 17y + 4\)
- Answer
-
Cubic
\(x = 9\)
- Answer
-
Linear
\(y = 2x + 1\)
- Answer
-
Linear
\(3y = 9x^2\)
- Answer
-
Quadratic
\(x^2 - 9 = 0\)
- Answer
-
Quadratic
\(y = x\)
- Answer
-
Linear
\(5x^7 = 3x^5 - 2x^8 + 11x - 9\)
- Answer
-
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.
\(5x+7\)
- Answer
-
binomial; first (linear); 5,7
\(16x + 21\)
\(4x^2 + 9\)
- Answer
-
binomial; second (quadratic); 4,9
\(7y^3 + 8\)
\(a^4 + 1\)
- Answer
-
binomial; fourth; 1,1
\(2b^5 - 8\)
\(5x\)
- Answer
-
monomial; first (linear); 5
\(7a\)
\(5x^3 + 2x + 3\)
- Answer
-
trinomial; third (cubic); 5,2,3
\(17y^4 + y^5 - 9\)
\(41a^3 + 22a^2 + a\)
- Answer
-
trinomial; third (cubic); 41,22,1
\(6y^2 + 9\)
\(2c^6 + 0\)
- Answer
-
monomial; sixth; 2
\(8x^2 - 0\)
\(9g\)
- Answer
-
monomial; first (linear); 9
\(5xy + 3x\)
\(3yz - 6y + 11\)
- Answer
-
trinomial; second (quadratic); 3,−6,11
\(7ab^2c^2 + 2a^2b^3c^5 + a^{14}\)
\(x^4y^3z^2 + 9z\)
- Answer
-
binomial; ninth; 1,9
\(5a^3b\)
\(6 + 3x^2y^5b\)
- Answer
-
binomial; eighth; 6,3
\(-9 + 3x^2 + 2xy6z^2\)
\(5\)
- Answer
-
monomial; zero; 5
\(3x^2y^0z^4 + 12z^3, y \not = 0\)
\(4xy^3z^5w^0, w \not = 0\)
- Answer
-
monomial; ninth; 4
Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it.
\(4x + 7 = 0\)
\(3y - 15 = 9\)
- Answer
-
linear
\(y = 5s + 6\)
\(y = x^2 + 2\)
- Answer
-
quadratic
\(4y = 8x + 24\)
\(9z = 12x - 18\)
- Answer
-
linear
\(y^2 + 3 = 2y - 6\)
\(y - 5 + y^3 = 3y^2 + 2\)
- Answer
-
cubic
\(x^2 + x - 4 = 7x^2 - 2x + 9\)
\(2y + 5x - 3 + 4xy = 5xy + 2y\)
- Answer
-
quadratic
\(3x - 7y = 9\)
\(8a + 2b = 4b - 8\)
- Answer
-
linear
\(2x^5 - 8x^2 + 9x + 4 = 12x^4 + 3x^3 + 4x^2 + 1\)
\(x - y = 0\)
- Answer
-
linear
\(x^2 - 25 = 0\)
\(x^3 - 64 = 0\)
- Answer
-
cubic
\(x^{12} - y^{12} = 0\)
\(x + 3x^5 = x + 2x^5\)
- Answer
-
fifth degree
\(3x^2 + 2x - 8y = 14\)
\(10a^2b^3c^6e^4 + 27a^3b^2b^4b^3b^2c^5 = 1, d \not = 0\)
- Answer
-
19th degree
The expression \(\dfrac{4x^3}{9x-7}\) is not a polynomial because.
The expression \(\dfrac{a^4}{7-a}\) is not a polynomial because.
- Answer
-
. . . there is a variable in the denominator
Is every algebraic expression a polynomial expression? If not, give an example of an algebraic expression that is not a polynomial expression.
Is every polynomial expression an algebraic expression? If not, give an example of a polynomial expression that is not an algebraic expression.
- Answer
-
Yes
How do we find the degree of a term that contains more than one variable?
Exercises for Review
Use algebraic notation to write “eleven minus three times a number is five.”
- Answer
-
\(11 - 3x = 5\)
Simplify \((x^4y^2z^3)^5\).
Find the value of \(z\) is \(z = \dfrac{x-u}{s}\) and \(x = 55, u = 49\) and \(s = 3\).
- Answer
-
\(z = 2\).
List, if any should appear, the common factors in the expression \(3x^4 + 6x^3 - 18x^2\).
State (by writing it) the relationship being expressed by the equation \(y = 3x + 5\).
- Answer
-
The value of \(y\) is \(5\) more than three times the value of \(x\).