What is Degree of Polynomial?
We have studied algebraic expressions and polynomials. To recall an algebraic expression f(x) of the form f(x) = a0 + a1x + a2x2 + a3 x3 + ……………+ an xn, there a1, a2, a3…..an are real numbers and all the index of ‘x’ are non-negative integers is called a polynomial in x.Polynomial comes from “poly” meaning "many" and “nominal” meaning "term" combinedly it means "many terms"A polynomial can have constants, variables, and exponents.
The degree of a polynomial is nothing but the highest degree of its exponent(variable) with a non-zero coefficient. Here the term degree means power. In this article let us study various degrees of polynomials.
What is the Degree of a Polynomial?
The highest degree exponent term in a polynomial is known as its degree.
To find the degree all that you have to do is find the largest exponent in the given polynomial.
For example, in the following equation:
f(x) = x3 + 2x2 + 4x + 3. The degree of the equation is 3 .i.e. the highest power of the variable in the polynomial is said to be the degree of the polynomial.
f(x) = 7x2 - 3x + 12 is a polynomial of degree 2.
thus,f(x) = an xn + an-1 xn-1 + an-2xn-2 +...................+ a1 x + a0 where a0 , a1 , a2 …....an are constants and an ≠ 0 .
Based on the degree of a polynomial, we have the following names for the degree of the polynomial.
Degree of Zero Polynomial
If all the coefficients of a polynomial are zero we get a zero degree polynomial. Any non - zero number (constant) is said to be zero degree polynomial if f(x) = a as f(x) = ax0 where a ≠ 0 .The degree of zero polynomial is undefined because f(x) = 0, g(x) = 0x , h(x) = 0x2 etc. are equal to zero polynomial.
Constant Polynomial
A polynomial having its highest degree zero is called a constant polynomial. It has no variables, only constants.
For example: f(x) = 6, g(x) = -22 , h(y) = 5/2 etc are constant polynomials. In general f(x) = c is a constant polynomial.The constant polynomial 0 or f(x) = 0 is called the zero polynomial.
Linear Polynomials
A polynomial having its highest degree Aries is called a linear polynomial.
For example, f(x) = x- 12, g(x) = 12 x , h(x) = -7x + 8 are linear polynomials.
In general g(x) = ax + b , a ≠ 0 is a linear polynomial.
Quadratic Polynomial
A polynomial having its highest degree 2 is known as a quadratic polynomial.
For example, f (x) = 2x2 - 3x + 15, g(y) = 3/2 y2 - 4y + 11 are quadratic polynomials.
In general g(x) = ax2 + bx + c, a ≠ 0 is a quadratic polynomial.
Cubic Polynomial
A polynomial having its highest degree 3 is known as a Cubic polynomial.
For example, f (x) = 8x3 + 2x2 - 3x + 15, g(y) = y3 - 4y + 11 are cubic polynomials.
In general g(x) = ax3 + bx2 + cx + d, a ≠ 0 is a quadratic polynomial.
Bi-quadratic Polynomial
A polynomial having its highest degree 4 is known as a Bi-quadratic polynomial.
For example, f (x) = 10x4 + 5x3 + 2x2 - 3x + 15, g(y) = 3y4 + 7y + 9 are quadratic polynomials.
In general g(x) = ax4 + bx2 + cx2 + dx + e, a ≠ 0 is a bi-quadratic polynomial.
Based on the degree of the polynomial the polynomial are names and expressed as follows:
Types of Polynomials Based on their Degrees
How to Find the Degree of a Polynomial?
There are simple steps to find the degree of a polynomial they are as follows:
Example: Consider the polynomial 4x5+ 8x3+ 3x5 + 3x2 + 4 + 2x + 3
Step 1: Combine all the like terms variables
(4x5 + 3x5) + 8x3 + 3x2 + 2x + (4 + 3)
Step 2: Ignore all the coefficients and write only the variables with their powers.
x5 + x3 + x2 + x + x0
Step 3: Arrange the variable in descending order of their powers if their not in proper order.
x5 + x3 + x2 + x1 + x0
Step 4: Check which the largest power of the variable and that is the degree of the polynomial
x5 + 33 + x2 + x + x0 = 5
Solved Examples
1. What is the Degree of the Following Polynomial
i) 5x4 + 2x3 +3x + 4
Ans: degree is 4
ii)11x9 + 10x5 + 11
Ans: degree is 9
2. Find the Zeros of the Polynomial.
p(x) = 3x - 2
Solution:
3x - 2 = 0
3x = 2
x= ⅔
x = 2/3 is a zero of p(x) = 3x - 2
Quiz Time
1. Write the Degrees of Each of the Following Polynomials.
12-x + 2x3
4x3 + 2x2 + 3x + 7
2. Identify the Polynomial
p(x) = 2x2 - x + 1
h(x) = x4 + 3x3 + 2x2 + 3
What is the Importance of the Degree of a Polynomial?
The degree of a polynomial is important. It helps in finding whether the given polynomial expression is homogeneous or not. To find the homogeneity of a polynomial expression, you have to find the degree of each term of the polynomial. For example, 2x3 + 3xy2 + 5y3 is a multivariable polynomial. If you have to find the homogeneity of this polynomial, you have to find the degree of each term. If the degrees of all terms are equal then the polynomial is homogeneous. If the degrees of all terms are not equal, then the polynomial is not homogeneous. In the above example, the degree of all terms is 3, thus the given expression is homogeneous.
Applications of Degree of a Polynomial
Some applications of degree of a polynomial are given here:
It helps in finding the maximum number of solutions that a function can have.
It helps in finding the maximum number of times a function can cross the x-axis when graphed.
It helps in finding if the given polynomial is homogeneous or not
Important Points to Remember
Degree of a polynomial with one variable is the highest exponent value of the variable in the given polynomial.
Degree of a polynomial with more than one variable can be found by adding the exponents of each variable in the given terms, and then find which term has the highest degree. That is the degree of the polynomial.
Degree of a rational expression can be found by taking the degree of the numerator and subtracting the degree of the denominator.
The degree of a polynomial expression with a square root is taken as ½
Fun Facts
A linear polynomial has only one zero
If the degree of the polynomial is n; the largest number of zeros it has is also n.
FAQs on Degree of Polynomial
1. What is the importance of polynomials in our daily lives?
Polynomials are the most vital parts of arithmetic and algebraic mathematics. In day-to-day life people make use of polynomials in their everyday life for modeling various buildings and objects, they use them in industries, for construction, etc. They also use them in marketing, finance, and stocks.
2. Why are polynomials necessary to study in mathematics?
Polynomials are a basic part of mathematics and algebra. They have the key role to play in each and every field of mathematics to express numbers performing mathematical operations.They are also called as "building blocks" for expressing other mathematical operations for example; rational expressions.
3. What is the meaning of Polynomial Equations?
An equation formed with variables, exponents, and coefficients together with operations and an equal sign is generally termed as a polynomial equation. It has different exponents which means the higher one gives the degree of the equation. Generally, the polynomial equation is expressed in the form of an(xn) a n ( x n ).
4. Which algebraic expression is polynomial?
As it is known; a polynomial is an algebraic sum in which no variables appear in denominators or under radical signs, and all variables that do appear are raised only to positive-integer powers are described as an algebraic expression in the polynomial.For example; the trinomial –2xy–1+3√x–4 is not a polynomial; however, the trinomial 3x2y4+√2xy–12 is a polynomial in the variables x and y.
5. What is the difference between a polynomial and not a polynomial equation?
As it is said that the polynomials can be identified by noting which expressions contain only the operations of addition, subtraction, multiplication, and non-negative integer exponents whereas; the non-polynomial expressions will be the expressions that contain other operations. This explains the biggest difference between polynomials and not polynomials.
6. What are polynomials? Mention its different types.
Polynomial comes from the word “poly” meaning "many" and “nomial” meaning "term" together it means "many terms"
Polynomials are algebraic expressions that may comprise exponents, variables and constants which are added, subtracted or multiplied but not divided by a variable. Polynomials are of different types, they are monomial, binomial, and trinomial. A monomial is a polynomial having one term. A binomial is an algebraic expression with two, unlike terms. A trinomial is an algebraic expression with three, unlike terms.
These are polynomials:
3x
x − 2
3xyz + 3xy2z
512v5 + 99w5
7
Yes, "7" is also polynomial, one term is allowed, and it can be just a constant.
These are not polynomials
3xy-2 is not, because the exponent is "-2" which is a negative number.
7/(x+5) is not, because dividing by a variable is not allowed
1/x is not either
√y is not, because the exponent is "½" .
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7. Explain different types of Polynomials.
Types of Polynomials
Let us get familiar with the different types of polynomials. They are as follows:
Monomials –An algebraic expressions with one term is called monomial hence the name “Monomial. In other words, it is an expression that contains any count of like terms. For example, 2x + 4x + 9x is a monomial because when we add the like terms it results in 15x. Furthermore, 21x2y, 8pq etc are monomials because each of these expressions contains only one term.
Binomials – An algebraic expressions with two unlike terms, is called binomial hence the name “Bi”nomial. For example, 3x + 5x2 is binomial since it contains two unlike terms, that is, 3x and 5x2. Likewise, 12pq + 13p2q is a binomial.
Trinomials – An expressions with three unlike terms, is called as trinomials hence the name “Tri”nomial. For example- 3x + 6x2 – 2x3 is a trinomial. It is due to the presence of three, unlike terms, namely, 3x, 6x2 and 2x3. Likewise, 11pq + 4x2 –10 is a trinomial.
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