- Examples of degree of a polynomial
- Table 1. Examples of polynomials and their degrees
- Procedure for working with polynomials
- Order, reduce, and complete a polynomial
- Importance of the degree of a polynomial in addition and subtraction
- Solved exercises
- - Exercise resolved 1
- Solution
- - Exercise solved 2
- Solution
- References
The degree of a polynomial in a variable is given by the term that has the greatest exponent, and if the polynomial has two or more variables, then the degree is determined by the sum of the exponents of each term, the greater sum being the degree of the polynomial.
Let's see how to determine the degree of the polynomial in a practical way.
Figure 1. Einstein's famous equation for energy E is a monomial of absolute degree 1 for the variable mass, denoted by m, since the speed of light c is considered constant. Source: Piqsels.
Suppose the polynomial P (x) = -5x + 8x 3 + 7 - 4x 2. This polynomial is one variable, in this case it is the variable x. This polynomial consists of several terms, which are the following:
And now what is the exponent? The answer is 3. Therefore P (x) is a polynomial of degree 3.
If the polynomial in question has more than one variable, then the degree can be:
-Absolute
-In relation to a variable
The absolute degree is found as explained at the beginning: adding the exponents of each term and selecting the largest.
Instead, the degree of the polynomial with respect to one of the variables or letters is the largest value of the exponent that said letter has. The point will become clearer with the examples and solved exercises in the following sections.
Examples of degree of a polynomial
Polynomials can be classified by degree, and can be first degree, second degree, third degree and so on. For the example in Figure 1, energy is a first degree monomial for mass.
It is also important to note that the number of terms that a polynomial has is equal to degree plus 1. Thus:
-First degree polynomials have 2 terms: a 1 x + a o
-The second degree polynomial has 3 terms: a 2 x 2 + a 1 x + a o
-A third degree polynomial has 4 terms: a 3 x 3 + a 2 x 2 + a 1 x + a or
And so on. The careful reader will have noticed that the polynomials in the previous examples are written in decreasing form, that is, placing the term with the greatest degree first.
The following table shows various polynomials, both of one and of several variables and their respective absolute degrees:
Table 1. Examples of polynomials and their degrees
| Polynomial | Degree |
|---|---|
| 3x 4 + 5x 3 -2x + 3 | 4 |
| 7x 3 -2x 2 + 3x-6 | 3 |
| 6 | 0 |
| x-1 | one |
| x 5 -bx 4 + abx 3 + ab 3 x 2 | 6 |
| 3x 3 and 5 + 5x 2 and 4 - 7xy 2 + 6 | 8 |
The last two polynomials have more than one variable. Of these, the term with the highest absolute degree has been highlighted in bold, so that the reader can quickly check the degree. It is important to remember that when the variable does not have a written exponent, it is understood that said exponent is equal to 1.
For example, in the highlighted term ab 3 x 2 there are three variables, namely: a, b and x. In this term, a is raised to 1, that is:
a = a 1
Therefore ab 3 x 2 = a 1 b 3 x 2
Since the exponent of b is 3 and that of x is 2, it immediately follows that the degree of this term is:
1 + 3 + 2 = 6
Y is the absolute degree of the polynomial, since no other term has a higher degree.
Procedure for working with polynomials
When working with polynomials, it is important to pay attention to the degree of it, since first and before performing any operation, it is convenient to follow these steps, in which the degree provides very important information:
-Order the polynomial of preference in decreasing direction. Thus, the term with the highest degree is on the left and the term with the lowest degree is on the right.
-Reduce like terms, a procedure that consists in adding algebraically all the terms of the same variable and degree found in the expression.
-If necessary, the polynomials are completed, inserting terms whose coefficient is 0, in case there are missing terms with an exponent.
Order, reduce, and complete a polynomial
Given the polynomial P (x) = 6x 2 - 5x 4 - 2x + 3x + 7 + 2x 5 - 3x 3 + x 7 -12, it is asked to order it in descending order, reduce the like terms, if any, and complete the missing terms. if accurate.
The first thing to look for is the term with the largest exponent, which is the degree of the polynomial, which turns out to be:
x 7
Therefore P (x) is of degree 7. Then the polynomial is ordered, starting with this term on the left:
P (x) = x 7 + 2x 5 - 5x 4 - 3x 3 + 6x 2 - 2x + 3x + 7 -12
Now the like terms are reduced, which are the following: - 2x and 3x on the one hand. And 7 and -12 on the other. To reduce them, the coefficients are added algebraically and the variable is left unchanged (if the variable does not appear next to the coefficient, remember that x 0 = 1):
-2x + 3x = x
7 -12 = -5
Replace these results in P (x):
P (x) = x 7 + 2x 5 - 5x 4 - 3x 3 + 6x 2 + x -5
And finally the polynomial is examined to see if any exponent is missing and indeed, a term whose exponent is 6 is missing, therefore it is completed with zeros like this:
P (x) = x 7 + 0x 6 + 2x 5 - 5x 4 - 3x 3 + 6x 2 + x - 5
Now it is observed that the polynomial was left with 8 terms, since as said before, the number of terms is equal to degree + 1.
Importance of the degree of a polynomial in addition and subtraction
With polynomials you can perform addition and subtraction operations, in which only like terms are added or subtracted, which are those with the same variable and the same degree. If there are no like terms, the addition or subtraction is simply indicated.
Once the addition or subtraction has been carried out, the latter being the sum of the opposite, the degree of the resulting polynomial is always equal to or less than the degree of the polynomial adding the highest degree.
Solved exercises
- Exercise resolved 1
Find the following sum and determine its absolute degree:
a 3 - 8ax 2 + x 3 + 5a 2 x - 6ax 2 - x 3 + 3a 3 - 5a 2 x - x 3 + a 3 + 14ax 2 - x 3
Solution
It is a polynomial with two variables, so it is convenient to reduce the like terms:
a 3 - 8ax 2 + x 3 + 5a 2 x - 6ax 2 - x 3 + 3a 3 - 5a 2 x - x 3 + a 3 + 14ax 2 - x 3 =
= a 3 + 3a 3 + a 3 - 8ax 2 - 6ax 2 + 14ax 2 + 5a 2 x - 5a 2 x + x 3 - x 3 - x 3 - x 3 =
= 5a 3 - 2x 3
Both terms are of degree 3 in each variable. Therefore the absolute degree of the polynomial is 3.
- Exercise solved 2
Express the area of the following plane geometric figure as a polynomial (figure 2 left). What is the degree of the resulting polynomial?
Figure 2. On the left, the figure for the solved exercise 2 and on the right, the same figure decomposed into three areas whose expression is known. Source: F. Zapata.
Solution
Since it is an area, the resulting polynomial must be of degree 2 in the variable x. To determine a suitable expression for the area, the figure is decomposed into known areas:
The area of a rectangle and a triangle are respectively: base x height and base x height / 2
A 1 = x. 3x = 3x 2; A 2 = 5. x = 5x; A 3 = 5. (2x / 2) = 5x
Note: the base of the triangle is 3x - x = 2x and its height is 5.
Now the three expressions obtained are added, with this we have the area of the figure as a function of x:
3x 2 + 5x + 5x = 3x 2 + 10x
References
- Baldor, A. 1974. Elementary Algebra. Cultural Venezolana SA
- Jiménez, R. 2008. Algebra. Prentice Hall.
- Wikibooks. Polynomials. Recovered from: es. wikibooks.org.
- Wikipedia. Degree (polynomial). Recovered from: es.wikipedia.org.
- Zill, D. 1984. Algebra and Trigonometry. Mac Graw Hill.