Lesson DEGREE OF A POLYNOMIAL
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This lesson provides an overview of how to determine the degree of a polynomial. REFERENCES <a href = "http://www.purplemath.com/modules/polydefs.htm" target = "_blank">http://www.purplemath.com/modules/polydefs.htm</a> <a href = "http://www.themathpage.com/aprecalc/polynomial.htm" target = "_blank">http://www.themathpage.com/aprecalc/polynomial.htm</a> <a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut25_poly.htm" target = "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut25_poly.htm</a> <a href = "http://tutorial.math.lamar.edu/Classes/Alg/Polynomials.aspx" target = "_blank">http://tutorial.math.lamar.edu/Classes/Alg/Polynomials.aspx</a> <a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut6_poly.htm" target = "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut6_poly.htm</a> <a href = "http://www.math.utah.edu/online/1010/poly/" target = "_blank">http://www.math.utah.edu/online/1010/poly/</a> <a href = "http://www.mathhelpforum.com/math-help/pre-calculus/107372-distinguishing-between-polynomial-rational-function.html" target = "_blank">http://www.mathhelpforum.com/math-help/pre-calculus/107372-distinguishing-between-polynomial-rational-function.html</a> <a href = "http://en.wikipedia.org/wiki/Polynomial" target = "_blank">http://en.wikipedia.org/wiki/Polynomial</a> DEFINITION OF A POLYNOMIAL A polynomial is an expression that is composed of terms separated by a plus or a minus sign. There are specific requirements for an expression to be a polynomial that will be covered later in this lesson. Example of a polynomial: Let A, B, C, and D be terms in a polynomial. If the first term is positive, then the polynomial would be expressed as A +/- B +/- C +/- D. If the first term is negative, then the polynomial would be expressed as -A +/- B +/- C +/- D. DEFINITION OF A TERM A term is an expression composed of constants and variables that are multiplied by each other. Example of a term in a polynomial: Let a,b,c, and d be constants. Let w,x,y,z be variables. A term in a polynomial would be expressed as a*w*b*x*c*y*d*z The constants in the term are usually combined together to form one constant. If we let e = a*b*c*d, then the term of "a*w*b*x*c*y*d*z" would be expressed as "e*w*x*y*z". DEFINITION OF THE COEFFICIENT OF A TERM The coefficient of a term in a polynomial is a constant that is multiplied by the variables in that term. If there are no variables in the term, then the constant stands by itself and the term is called a constant. In the term of e*w*x*y*z, the letter e would be called the coefficient of that term. DEFINITION OF A CONSTANT A constant is a fixed value. It cannot change. It is usually denoted as a real number, such as 1, 2, 3, 5.5, 7.9, etc. It can also be expressed as a letter which is usually one of the lower letters of the alphabet, such as a,b,c,d,e, ... DEFINITION OF A VARIABLE A variable is a letter that represents a value that can change. It is usually represented by letters high in the alphabet, such as u,v,w,x,y,z ... VERY BRIEF REVIEW OF LAWS OF ARITHMETIC a * b = b * a (Commutative) a * (b * c) = (a * b) * c (Associative) a * 1 = a (Identity) If a * b = c, then b = c/a and a = c/b (Inverse) If a and b are real, then c = a * b is also real. a + b = b + a (Commutative) a + (b + c) = (a + b) + c (Associative) a + 0 = a (Identity) If a + b = c, then a = c - b and b = c - a (Inverse) If a and b are real, then c = a + b is also real. a * (b + c) = a * b + a * c (Distributive) (a + b) * (c + d) = a * (c + d) + b (c + d) = a*c + a*d + b*c + b*d (Distributive) Division is a form of Multiplication: a / b is the same as a * (1/b) Subtraction is a form of Addition: a - b = a + (-b) While Multiplication by 0 is allowed, Division by 0 is not allowed. The Order of arithmetic operations are from left to right. Expressions within the inner set of parentheses are resolved first. -------------------- Within each set of parentheses: Exponents and Radicals are resolved first Multiplication and Division are resolved next. Addition and Subtraction are resolved next. -------------------- Once all parentheses have been resolved, there is a final pass to complete the operations on the overall expression. DEGREE OF A CONSTANT The degree of a constant is always equal to 0. It doesn't matter how the constant is expressed. The degree of 5 is equal to 0. The degree of {{{5^2}}} is equal to 0. The degree of {{{sqrt(25)}}} is equal to 0. The degree of 5*7*9*16 is equal to 0. Any constant can be represented as that constant times a variable raised to the power of 0 because any variable raised to the power of 0 is always equal to 1. This variable is not shown but it is implied. {{{15 * x^0}}} = {{{15 * 1}}} = {{{15}}}. The degree of the constant, by itself, is really the degree of the implied variable it is multiplied by which is always equal to 0. The exponents of a constant do not count when determining the degree of a polynomial. It is only the exponents of the variables that count. DEGREE OF A VARIABLE The degree of a variable is the value of the exponent of that variable. The degree of {{{x^0}}} is 0 because the variable of x is raised to the power of 0. The value of {{{x^0}}} is always equal to 1, regardless of the value of x. The degree of x is 1 because x is really {{{x^1}}} which means that the variable of x is raised to the power of 1. The degree of {{{x^2}}} is 2 because the variable of x is raised to the power of 2. The degree of {{{y^2}}} is 2 because the variable of y is raised to the power of 2. DEGREE OF A TERM The degree of a term is equal to the sum of the exponents of the variables in that term. The exponents of any constants in the term do not count. It is only the exponents of the variables in a term that count. Using that rule, we'll determine the degree of the following terms: {{{16}}} has a degree of 0. ----- {{{16^5}}} has a degree of 0. ----- {{{16*15*4}}} has a degree of 0. ----- {{{root(7,16)}}} has a degree of 0. ----- {{{x^0}}} has a degree of 0. ----- {{{16*x^0}}} has a degree of 0. ----- {{{x}}} has a degree of 1. ----- {{{16*x}}} has a degree of 1. ----- {{{16*x^7}}} has a degree of 7. ----- {{{16*x^7*x^9}}} has a degree of 16. Note that {{{16*x^7*x^9}}} can be simplified to {{{16*x^16}}} by applying the law of exponentiation that states that {{{x^m*x^n = x^(m+n)}}}. ----- {{{5^32*x^7*y^9}}} has a degree of 16. Note that the law of exponentiation that states that {{{x^m*x^n = x^(m+n)}}} doesn't apply here because the variables in this term are not the same. Note also that the degree of the constant doesn't count, regardless of what power it is raised to. DEGREE OF A POLYNOMIAL The degree of a polynomial is equal to the degree of the term in the polynomial that has the highest degree. Example: {{{(x^7*y^6*z^5*14*12^3*sqrt(32)) + (3) - (w^13)}}} The first term in this polynomial is {{{x^7*y^6*z^5*14*12^3*sqrt(32)}}}. the second term in this polynomial is {{{3}}} The third term in this polynomial is {{{w^13}}} The degree of the first term is equal to 7 + 6 + 5 = 18 because that is the sum of the exponents of the variables in that term. The degree of the second term is equal to 0 because that is the sum of the exponents of the variables in that term. The degree of the third term is equal to 13 because that is the sum of the exponents of the variables in that term. The degree of the term is the sum of the exponents of the variables in that term only. The sum of the exponents of the constants in that term are ignored. The degree of this polynomial is equal to 18 because that is the degree of the term in the polynomial that has the highest degree. STANDARD FORM OF A POLYNOMIAL The standard form of a polynomial shows the term with the highest degree first, then the term with the next highest degree, then the term with the next highest degree, going from left to right. The standard form of the polynomial of {{{x^7*y^6*z^5*14*12^3*sqrt(32) + 3 - w^13}}} would be shown as: {{{x^7*y^6*z^5*14*12^3*sqrt(32) - w^13 + 3}}} Whether you have expressed the polynomial in the standard form or not has no impact on the degree of the polynomial. REQUIREMENTS FOR AN EXPRESSION TO BE CALLED A POLYNOMIAL As best I can determine, the requirements for an expression to be called a polynomial include the following criteria: 1. It must be continuous. 3. The exponents of any variables, or expressions that contain variables, have to be non-negative integers only. 4. The number of terms in the expression must be finite. The fact that the expression must be continuous means that there should be no points of discontinuity nor any points of abrupt changes in direction. The fact that the variables, or expressions that contain variables, must have exponents that are non-negative integers only means that no variables, or expressions that contain variables, can be in the denominator of any term. Based on these criteria, we can determine if the following expressions are polynomials. {{{1/x}}} is not a polynomial because the x is in the denominator of the term. <b>x<sup>(3/2)</sup></b> is not a polynomial because it has a rational exponent that does not resolve to an integer. {{{abs(x)}}} is not a polynomial because it has an abrupt change in direction at x = 0 as shown in the following graph: {{{graph(300,300,-5,5,-5,5,abs(x))}}} {{{5/(x+1)}}} is not a polynomial because the expression {{{(x+1)}}} is in the denominator of the term. The graph of this equation is shown below: <font color = "red"><img src = "http://theo.x10hosting.com/examples/Polynomials/Polynomials2.jpg" alt = "picture not found" /></font> Note that this function is not continuous. It has a point of discontinuity at x = -1. <b>x<sup>3</sup> * 15<sup>(-1)</sup> + 9<sup>(1/2)</sup></b> is a polynomial because all variables contain positive integer exponents. The exponent of the variable is a positive integer. The exponent of the constant in the first term is a negative integer but this doesn't count because the exponent is not associated with a variable. The exponent in the second term is rational that does not resolve to a non-negative integer but this doesn't count either because the exponent is associated with a constant and not a variable. A graph of this equation looks like this: {{{graph(300,300,-5,5,-20,20,x^3/15 + 9^.5)}}} <b>x<sup>(4/2)</sup></b> is a variable with a rational expression that resolves to a non-negative integer, so this expression is a polynomial because it resolves to <b>x<sup>2</sup></b>. SOME SPECIAL CASES OF EXPRESSIONS BEING OR NOT BEING POLYNOMIALS The reference for this section is <a href = "http://www.math.utah.edu/online/1010/poly/" target = "_blank">http://www.math.utah.edu/online/1010/poly/</a> While {{{y = abs(x)}}} is not a polynomial, {{{y = abs(x)^2}}} is a polynomial. The graph of {{{y = abs(x)^2}}} is shown below: {{{graph(300,300,-5,5,-5,5,abs(x)^2)}}} You can see that this graph is continuous, while the graph of {{{y = abs(x)}}} is not. {{{y = (x^4-1)/(x^2+1)}}} looks like it should not be a polynomial because the expression {{{(x^2+1)}}} is in the denominator. When you simplify this expression, however, you get: {{{((x^2+1)*(x^2-1))/(x^2+1)}}} which simplifies to {{{x^2-1}}} which is a polynomial. Notice that there are no points of discontinuity in this equation because the expression in the denominator of {{{x^2+1}}} is always positive, regardless if x is negative or positive. A graph of the equation {{{y = (x^4-1)/(x^2+1)}}} looks like this: {{{graph(300,300,-5,5,-5,5,(x^4-1)/(x^2+1))}}} A graph of the equation {{{y = x^2 - 1}}} looks the same: {{{graph(300,300,-5,5,-5,5,x^2-1)}}} The expression of {{{(x^2-1)/(x-1)}}}, however, is not a polynomial, even though {{{(x^2 - 1)/(x-1)}}} is equivalent to {{{((x-1)*(x+1))/(x-1)}}} which simplifies to {{{(x+1)}}}. The reason for this is that the denominator becomes 0 when x = 1 which introduces a point of discontinuity in the expression. Once you simplify the expression, it looks continuous, but this simplification applies to all values of x except x = 1. The expression is undefined when x = 1. The graph of the equation of {{{y = (x^2-1)/(x-1)}}} is shown below: <font color = "red"><img src = "http://theo.x10hosting.com/examples/Polynomials/Polynomials3.jpg" alt = "picture not found" /></font> You can see that the graph looks continuous except where x = 1. There is a hole there as the value of y skyrockets to infinity. Since infinity is not a value, the value of y when x = 1 is undefined. The graph of the equation of {{{y = (x+1)}}} is shown below: <font color = "red"><img src = "http://theo.x10hosting.com/examples/Polynomials/Polynomials4.jpg" alt = "picture not found" /></font> This graph is continuous, but the simplification from the equation of {{{y = (x^2-1)/(x-1)}}} to the equation of {{{y = (x+1)}}} is only valid when x is not equal to 1. Based on the reference this came from, if you define y as being equal to 2 when x equals 1, then the equation of {{{y = (x^2-1)/(x-1)}}} can be defined as continuous, but a strict definition of a continuous function would not allow that. We have an expression that looks like it could be a polynomial, but is not, if we follow the strict definition because of the discontinuity at x = 1, although we can still call it a polynomial if we allow the definition of y = 2 when x = 1 to be included. GRAPHING SOFTWARE A word of caution about graphing software. Not all graphing software will pick up the discontinuity. Here's an example of one that didn't. <font color = "red"><img src = "http://theo.x10hosting.com/examples/Polynomials/Polynomials5.jpg" alt = "picture not found" /></font> While this graphing software did not pick up the discontinuity, the table associated with the graph did. Look at the right where the value of x is equal to 1 and you will see that the value of y is left blank. You cannot rely on the graphing software to show you the points of discontinuity. Some will show it and some will not. You have to look at the original equation and determine where the points of discontinuity exist, if there are any. SUMMARY Some time was spent in determining the nature of a polynomial because all expressions are not polynomials. Some additional time was spent in clarifying the definitions of some of the terms used, and in reviewing some of the arithmetic operations that apply to polynomial expressions. The main thrust of this lesson, however, is determining the degree of the polynomial. After all is said and done, determining the degree of a polynomial can be summarized into the following two statements: 1. The degree of a term in a polynomial is the sum of the exponents of the variables in that term. 2. The degree of a polynomial is equal to the degree of the term in the polynomial that has the highest degree.
