Degree of a Polynomial
Polynomials are one of the significant concepts of mathematics, and so is the degree of polynomials, which determines the maximum number of solutions a function could have and the number of times a function will cross the x-axis when graphed. It is the highest exponential power in the polynomial equation. Let's learn in detail about the degree of a polynomial and how to find the degree of a polynomial.
|1.||What is Degree of a Polynomial?|
|2.||How to Find the Degree of a Polynomial?|
|3.||Polynomials Based on Degree|
|4.||Degree of a Polynomial Applications|
What is Degree of a Polynomial?
The degree of a polynomial is the highest exponential power in the polynomial equation. Only variables are considered to check for the degree of any polynomial, coefficients are to be ignored. For an nth degree polynomial function with real coefficients and the variable is represented as x, having the highest power n, where n takes whole number values.
Degree of a Polynomial Definition:
Degree of a polynomial is the greatest power of a variable in the polynomial equation. To determine the degree of a polynomial function, only terms with variables are considered to find out the degree of any polynomial. The highest exponential power of the variable term in the polynomial indicates the degree of that polynomial.
Look at the polynomial function given below, where the highest power of x is n. Hence, n is the degree of polynomial in this function.
How to Find the Degree of a Polynomial?
Consider the polynomial: p(x):2x5−12x3+3x−π. The term with the highest power of x is 2x5 and the corresponding (highest) exponent is 5. Therefore, we will say that the degree of this polynomial is 5. Thus, the degree of a polynomial is the highest power of the variable in the polynomial. We can represent the degree of a polynomial by Deg(p(x)). Given below are some examples:
Note from the last example above that the degree is the highest exponent of the variable term, so even though the exponent of π is 3, that is irrelevant to the degree of the polynomial.
Degree of a Zero Polynomial
When all the coefficients are equal to zero, the polynomial is considered to be a zero polynomial. So, the degree of the zero polynomial is either undefined or defined in a way that is negative (-1 or ∞).
Degree of a Constant Polynomial
A constant polynomial (P(x) = c) has no variables. Since there is no exponent so no power to it. Thus, the degree of the constant polynomial is zero. For example: For 6 or 6x0, degree = 0.
Degree of a Polynomial With More Than One Variable
The degree of a polynomial with more than one variable can be calculated by adding the exponents of each variable in it. For example: 5x3 + 6x2y2 + 2xy
- 5x3 has a degree of 3 (x has an exponent of 3)
- 6x2y2 has a degree of 4 (x has an exponent of 2, y has 2, and 2+2=4)
- 2xy has a degree of 2 (x has an exponent of 1, y has 1, and 1+1=2)
The largest degree out of those is 4, so the polynomial has a degree of 4.
Polynomials Based on Degree
Each of the polynomials has a specific degree and based on that they have been assigned a specific name. Let's classify the polynomials based on the degree of a polynomial with examples.
|Constant Polynomial||Polynomials with Degree 0||3|
|Linear Polynomial||Polynomials with Degree 1||x + 8|
|Quadratic Polynomial||Polynomials with Degree 2||3x2 - 4x + 7|
|Cubic Polynomial||Polynomials with Degree 3||2x3 + 3x2 + 4x + 6|
|Quartic Polynomial||Polynomials with Degree 4||x4-16|
|Quintic Polynomial||Polynomials with Degree 5||4x5+ 2x3 - 20|
Degree of a Polynomial Applications
Given below are a few applications of the degree of a polynomial:
- To determine the most number of solutions that a function could have.
- To determine the most number of times a function will cross the x-axis when graphed.
- To check whether the polynomial expression is homogeneous, determine the degree of each term. When the degrees of the term are equal, then the polynomial expression is homogeneous and when the degrees are not equal, then the expression is said to be non-homogenous.
4x3 + 3xy2+8y3
The degree of all the terms is 3. Hence, the given example is a homogeneous polynomial of degree 3.
Tips and Tricks
In order to find the degree of any polynomial, you can follow these steps:
- Identify each term of the given polynomial.
- Combine all the like terms, the variable terms; ignore constant terms.
- Arrange those terms in descending order of their powers.
- Find the term with the highest exponent and that defines the degree of the polynomial.
- Degree of a polynomial with only one variable: The largest exponent of the variable in the polynomial.
- Degree of a polynomial with more than one variable: To find the degree of the polynomial, you first have to identify each term of that polynomial, so to find the degree of each term you add the exponents.
- Degree of a rational expression: Take the degree of the top (numerator) and subtract the degree of the bottom (denominator).
- Degree of any polynomial expression with a root such as 3√x is 1/2.
Solved Examples on Degree of a Polynomial
Example 1: Determine the degree and the leading coefficient of the following polynomial expression 5x2 - 20x - 20.
Given polynomial expression, 5x2 - 20x - 20. The highest exponent is 2, and so the degree of the expression is 2. The coefficient with the highest exponent will be the leading coefficient of the expression, so the leading coefficient is 5.
Therefore, degree= 2 and leading coefficient= 5.
Example 2: Find the degree of the polynomial 5x4 + 3x2 - 7x5 + x7
In order to find the degree of the given polynomial, Check each term of the given polynomial. All are like terms with x as a variable. Arrange these terms in descending order of their powers, which gives x7 - 7x5 + 5x4+ 3x2 Term with the greatest or highest exponent is x7, so the degree of the polynomial is 7Therefore, the degree of the polynomial is 7.
Example 3: Find a fourth-degree polynomial satisfying the following conditions: has roots- (x-2), (x+5) that is divisible by 4x2
We are already familiar with the fact that a fourth degree polynomial is a polynomial with degree 4. Also, we know that we can find a polynomial expression by its roots. First condition: (x-2) (x+5) = x(x+5) - 2(x+5) = x2+5x-2x-10 = x2+3x-10. Second condition: (x2+3x-10)(4x2) = x2.4x2 + 3x.4x2 - 10.4x2 = 4x4+12x3-40x2
Therefore, the required polynomial = 4x4 + 12x3- 40x2
FAQs on Degree of a Polynomial
What is the Degree of a Polynomial?
The degree of a polynomial is the highest degree of the variable term, with a non-zero coefficient, in the polynomial.
What is the Degree of a Quadratic Polynomial?
Quadratic Polynomials are characterized as polynomials with degree 2. Thus, the degree of a quadratic polynomial is 2.
What is Degree 3 Polynomial?
Degree 3 polynomials have one to three roots, two or zero extrema, one inflection point with a point symmetry about the inflection point, roots solvable by radicals, and most importantly degree 3 polynomials are known as cubic polynomials.
What is the Degree of the Zero Polynomial?
A Zero Polynomial has all its variable coefficients equal to zero. It is a constant polynomial having a value 0. Thus, the degree of the zero polynomial is undefined.
What is the Degree of Polynomial 5√x?
For the polynomial 5√x, the exponent with variable x is 1/2. Thus, the degree of 5√x is 1/2.
What is the Degree of the Polynomial 5 √ 3?
The degree of the polynomial 5 √ 3 is zero as there is no variable and the degree of any polynomial is defined by the highest exponential power of its variable term.
Why is the Degree of a Polynomial Important?
The degree of a polynomial function has great importance as it determines the maximum number of solutions that a function could have and the maximum number of times a function crosses the x-axis on graphing it.