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 3x 4 Example 2 2 5 x2 y 6. A fraction occurs, but no variable appears in the denominator. * Version 1.4: May 28, 2009 8:42 pm +0000 http://creativecommons.org/licenses/by/3.0/
OpenStax-CNX module: m21848 2 Example 3 5x 3 + 3x 2 2x + 1 Some expressions that are not polynomials are Example 4 3 x 16. A variable appears in the denominator. Example 5 4x 2 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 Comment One Monomial 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. 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 5x 3 is a monomial of degree 3. Example 7 60a 5 is a monomial of degree 5. Example 8 21b 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 8x 0. Since x 0 = 1 (x 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.) Example 10 4x is a monomial of the rst degree. 4xcould be written as 4x 1. The exponent on the variable is 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 is the sum of the exponents of the variables, as shown below.
OpenStax-CNX module: m21848 3 Example 11 4x 2 y 5 is a monomial of degree 2 + 5 = 7. This is a 7th degree monomial. Example 12 37ab 2 c 6 d 3 is a monomial of degree 1 + 2 + 6 + 3 = 12. This is a 12th degree monomial. Example 13 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: Example 14 2x 3 + 6x 1 is a trinomial of degree 3. The rst term, 2x 3, 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 + 5a 2 is a trinomial of degree 2. Example 17 2x 6 + 9x 4 x 7 8x 3 + x 9 is a polynomial of degree 7. Example 18 4x 3 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 Polynomials of the rst degree are called polynomials. Polynomials of the second degree are called 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. Some examples of these polynomials follow: Example 20 4x 9 is a polynomial. Example 21 3x 2 + 5x 7 is a polynomial. Example 22 8y 2x 3 is a cubic polynomial. Example 23 16a 2 32a 5 64 is a 5th degree polynomial. Example 24 x 12 y 12 is a 12th degree polynomial. Example 25 7x 5 y 7 z 3 2x 4 y 7 z + x 3 y 7 is a 15th degree polynomial. The rst term is of degree 5 + 7 + 3 = 15. Example 26 43 is a 0th degree polynomial.
OpenStax-CNX module: m21848 4 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 equation since it is of degree 1, the degree of the expression on the left of the = sign. Example 28 5x 2 + 2x 7 = 4 is a equation since it is of degree 2. Example 29 9x 3 8 = 5x 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 is a 4th degree equation. Example 31 a 5 3a 4 = a 3 + 6a 4 7 is a 5th degree equation. Example 32 y = 2 3x + 3 is a equation. Example 33 y = 3x 2 1 is a equation. Example 34 x 2 y 2 4 = 0 is a 4th degree equation. The degree of x 2 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.) 9x 2 + 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 (Solution on p. 8.) 3y = 9x 2
OpenStax-CNX module: m21848 5 Exercise 7 (Solution on p. 8.) x 2 9 = 0 Exercise 8 (Solution on p. 8.) y = x Exercise 9 (Solution on p. 8.) 5x 7 = 3x 5 2x 8 + 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.) 4x 2 + 9 Exercise 13 7y 3 + 8 Exercise 14 (Solution on p. 8.) a 4 + 1 Exercise 15 2b 5 8 Exercise 16 (Solution on p. 8.) 5x Exercise 17 7a Exercise 18 (Solution on p. 8.) 5x 3 + 2x + 3 Exercise 19 17y 4 + y 5 9 Exercise 20 (Solution on p. 8.) 41a 3 + 22a 2 + a Exercise 21 6y 2 + 9 Exercise 22 (Solution on p. 8.) 2c 6 + 0 Exercise 23 8x 2 0 Exercise 24 (Solution on p. 8.) 9g Exercise 25 5xy + 3x Exercise 26 (Solution on p. 8.) 3yz 6y + 11
OpenStax-CNX module: m21848 6 Exercise 27 7ab 2 c 2 + 2a 2 b 3 c 5 + a 14 Exercise 28 (Solution on p. 8.) x 4 y 3 z 2 + 9z Exercise 29 5a 3 b Exercise 30 (Solution on p. 8.) 6 + 3x 2 y 5 b Exercise 31 9 + 3x 2 + 2xy6z 2 Exercise 32 (Solution on p. 8.) 5 Exercise 33 3x 2 y 0 z 4 + 12z 3, y 0 Exercise 34 (Solution on p. 8.) 4xy 3 z 5 w 0, w 0 Classify each of the equations for the following problems by degree. If the term,, 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 = x 2 + 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 x 2 + x 4 = 7x 2 2x + 9 Exercise 44 (Solution on p. 9.) 2y + 5x 3 + 4xy = 5xy + 2y Exercise 45 3x 7y = 9 Exercise 46 (Solution on p. 9.) 8a + 2b = 4b 8 Exercise 47 2x 5 8x 2 + 9x + 4 = 12x 4 + 3x 3 + 4x 2 + 1
OpenStax-CNX module: m21848 7 Exercise 48 (Solution on p. 9.) x y = 0 Exercise 49 x 2 25 = 0 Exercise 50 (Solution on p. 9.) x 3 64 = 0 Exercise 51 x 12 y 12 = 0 Exercise 52 (Solution on p. 9.) x + 3x 5 = x + 2x 5 Exercise 53 3x 2 y 4 + 2x 8y = 14 Exercise 54 (Solution on p. 9.) 10a 2 b 3 c 6 d 0 e 4 + 27a 3 b 2 b 4 b 3 b 2 c 5 = 1, d 0 Exercise 55 The expression 4x3 9x 7is not a polynomial because Exercise 56 (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 (Solution on p. 9.) () Use algebraic notation to write eleven minus three times a number is ve. Exercise 61 () Simplify ( x 4 y 2 z 3) 5. Exercise 62 (Solution on p. 9.) () Find the value of zif z = x u s and x = 55, u = 49,and s = 3. Exercise 63 () List, if any should appear, the common factors in the expression 3x 4 + 6x 3 18x 2. Exercise 64 (Solution on p. 9.) () State (by writing it) the relationship being expressed by the equation y = 3x + 5.
OpenStax-CNX module: m21848 8 Solutions to Exercises in this Module rst, or cubic eighth degree binomial; rst (); 5, 7 binomial; second (); 4, 9 binomial; fourth; 1, 1 monomial; rst (); 5 trinomial; third (cubic); 5, 2, 3 trinomial; third (cubic); 41, 22, 1 monomial; sixth; 2 monomial; rst (); 9 trinomial; second (); 3, 6, 11 binomial; ninth; 1, 9 binomial; eighth; 6, 3 monomial; zero; 5 monomial; ninth; 4
OpenStax-CNX module: m21848 9 cubic cubic fth degree 19th degree... there is a variable in the denominator yes 11 3x = 5 z = 2 The value of y is 5 more then three times the value of x.