Degree of Polynomials

Before discussing the degree of polynomials let us first discuss what a polynomial is:

What is a Polynomial?

Let \(x\) be a variable, \(n\) be a positive integer and \({a_0},\,{a_1},\,{a_2}, \ldots ,\,{a_n}\) be constants (real numbers). Then, \(f(x) = {a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} +  \ldots  + {a_1}x + {a_0}\) is known as a polynomial in variable \(x\) with degree \(n\).

Examples: \(7x + 3,\,11y – 6\) are some examples of polynomials.

Degree of the Polynomial: Definition

A polynomial’s degree is the highest power of a variable or highest exponential power in a given polynomial equation (ignoring the coefficients).
For instance: Consider the polynomial 5x⁴ + 7x³+ 9l.  Here, the terms in the polynomial are 5x⁴, 7x³, 9, where 5x⁴ is the term with the highest power i.e. 4. The coefficients of the polynomial are 5 and 7.
The degree of the polynomial 5x⁴ + 7x³+ 9 is 4.

Similarly,
(f(x) = 3x + \frac{1}{2}) is a polynomial of degree (1) in the variable (x).
(g(y) = 2{y^2} – \frac{3}{2}y + 7) is a polynomial of degree (2) in the variable (y).
(p(x) = 5{x^3} – 3{x^2} + x – \frac{1}{{\sqrt 2 }}) is a polynomial of degree (3) in a variable (x).

Degree of Multivariable Polynomial

Polynomials can be classified into polynomial with one variable (univariate polynomial) and polynomial with multiple variables (multivariable polynomial).  We already mentioned that the degree of the polynomial with one variable is the highest power of the polynomial expression. However, if a polynomial has more than one variable, the degree of the polynomial is found by summing up the powers of all the variables in any term of the polynomial. 

Let’s consider a polynomial expression with two variables, say x and y

(i.e) 2x³ + 4x³y⁴ + 5y²+3

The degree of this polynomial is 7 because, in the second term,  4x³y⁴, the exponent values of x is 3 and y is 4. When we add the exponent values, we get 7. Thus, the degree of the above multivariable polynomial expression is 7. Thus, if “a” and “b” are the exponents or the powers of the variable, then the degree of the polynomial is “a + b”, where “a” and “b” are the whole numbers.

Why is the Degree of Any Constant Term Zero?

A constant term is a term in an algebraic expression that has a value that does not change in the given condition.
In the constant term, the power of a variable is zero. Since \(17\) can be written as \(17 \times 1 = 17 \times {y^{\rm{o}}}\)  (we associate the variable \(y\) with power \(0\) with the constant term \(17\)) [\(y\) to the power\(0\) is \(1\)].
So, every constant term may be thought of as associated with a variable with power \(0\). Hence the degree of any constant term is \(0\).

Types of Polynomials

Based on the number of the terms they contain:

1. Monomial: It is an algebraic expression that contains only one term.
   Example: \(5x,\,2xy,\, – 3{a^2}b,\, – 7\) etc., are monomials.
2. Binomials: An algebraic expression that contains two terms is known as binomial.
   Example: \((2a + 3b),\,(8 – 3x),\,\left( {{x^2} – 4x{y^2}} \right)\) etc., are binomials.
3. Trinomials: An algebraic expression containing three terms is known as trinomial.
   Example: \((a + 2b + 5c),\,(x + 2y – 3z),\,\left( {{x^3} – {y^3} – {z^3}} \right)\) etc., are trinomials.
4. Quadrinomials: An algebraic expression containing four terms is known as a quadrinomial.
   Examples: \((x + y + z – 5),\,\left( {{x^3} + {y^3} + {z^3} + 3xyz} \right)\), etc., are quadrinomials.
5. Polynomials: A expression containing two or more terms is known as a polynomial. It includes Binomials, Trinomials, Quadrinomials and all the algebraic expressions with five or more terms.

Calssification of Polynomials Based on the Highest Degree of Terms

Polynomials are classified into first degree, second degree, third degree etc. depending on the highest degree of the variable of the polynomial.

First Degree (Linear Polynomial)

A polynomial of degree \(1\) is known as a linear polynomial.
For example:
\(p(x) = 4x – 3,\,q(y) = 3y,\,f(t) = 3t + 5\) and \(g(u) = \frac{2}{3}u – \frac{5}{2}\) etc., are all linear polynomials.
Polynomials such as \(f(x) = 2{x^2} + 3,\,g(x) = 3 – {x^2}\) etc., are not linear polynomials.
More generally, any linear polynomial in variable \(x\) with real coefficients is of the form \(f(x) = ax + b\) where \(a,\,b\) are real numbers and \(a \ne 0\).
Remark: A linear polynomial may be a monomial or a binomial because linear polynomial \(f(x) = \frac{2}{3}x – \frac{5}{2}\) is a binomial whereas the linear polynomial \(g(x) = \frac{2}{5}x\) is a monomial.

Second Degree (Quadratic Polynomial)

A polynomial of degree \(2\) is called a quadratic polynomial. The name ‘quadratic’ has been derived from ‘quadrate’ which means, ‘square’.
For example:
\(f(x) = 2{x^2} + 3x – \frac{4}{5},\,g(y) = 2{y^2} – 3,\,h(u) = 2 – {u^2} + \sqrt 3 u,\,p(v) = \sqrt 3 {v^2} – \frac{4}{3}v + \frac{1}{2}\)
\(q(a) = \frac{2}{3}{a^2} + 4a\) etc., are quadratic polynomials with real coefficient.
More generally, any quadratic polynomial in variable \(x\) with real coefficient is of the form \(f(x) = a{x^2} + bx + c\) where \(a,\,b,\,c\) are real numbers and \(a \ne 0\).
Remark: A quadratic polynomial may be a monomial or a binomial, or a trinomial because \(f(x) = \frac{1}{5}{x^2}\) is a monomial, \(g(x) = 3{x^2} – 5\) is binomial and \(h(x) = 3{x^2} – 2x + 5\) is a trinomial.

Third Degree (Cubic Polynomial)

A polynomial of degree \(3\) is known as a cubic polynomial.
For example:
(i). \(f(x) = \frac{9}{5}{x^3} – 2{x^2} + \frac{7}{3}x – \frac{1}{5}\) is a cubic polynomial in variable \(x\).
(ii). \(g(y) = 2{y^3} + 5y – 7\) is a cubic polynomial in variable \(y\).
(iii). \(p(u) = \frac{{\sqrt 2 }}{3}{u^3} + 1\) is a cubic polynomial in variable \(u\).

The most general form of a cubic polynomial with coefficients as real numbers is \(f(x) = a{x^3} + b{x^2} + cx + d\) where \(a \ne 0,\,b,\,c,\,d\) are real numbers.

Special Polynomials

Some of the special polynomials are:

Zero Polynomial

Definition: We learnt earlier that \(f(x) = {a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} +  \ldots  + {a_1}x + {a_0}\) is a polynomial. Now, a constant polynomial is a special case of the polynomial where all the coefficients \({a_n},\,{a_{n – 1}},\,{a_{n – 2}},\, \ldots  \ldots {a_1},\,{a_0}\) are zeros \((0)\) Hence, the polynomial becomes \(f(x) = 0\) or the corresponding polynomial function is the constant function with value \(0\).
\(f(x) = 0,\,g(x) = 0x,\,h(x) = 0{x^2},\,p(x) = 0{x^3},\,q(x) = 0{x^{12}}\) etc. are all examples of zero polynomials.
The zero polynomial is considered as the additive identity of the polynomials.
The degree of a zero polynomial is not defined.

Constant Polynomial

Definition: A polynomial of degree zero is called a constant polynomial.
For example \(f(x) = 7,\,g(x) =  – \frac{3}{2},\,h(y) = 2,\,p(t) = 1\) etc., are constant polynomials.
Degree of Constant Polynomial: The constant polynomial’s value remains the same; it contains no variable. For example: In \(p(x) = c\), c is a constant. No variable is associated with it. Thus, there is no exponent, and hence, no power to the exponent. So, the power of the constant polynomial is \(0\). In other words, any of the constant can be written with the variable with the exponential power of \(0\). For example, the constant term \(6\) can be written in a polynomial form as \(p(x) = 6{x^0}\).

Importance of Degree of Polynomial

1. The concept of the degree of polynomials can also be applied to the degree of the equations. If we write the polynomial \(f(x)\) as \(f(x) = 0\), it is called a polynomial equation. Hence, the degree of the polynomial is the same as the degree of the polynomial equation or simply the equation.
2. The number of zeros or solutions of a polynomial is equal to its degree. A first-degree equation or a linear polynomial equation has one solution (one zero). A second-degree equation or a quadratic polynomial equation has two solutions (two zeros). A third-degree equation or a cubic polynomial equation has three solutions (three zeros).
3. Given the degree of a polynomial, we can understand the nature or shape of its graph immediately.

The first-degree or a linear polynomial has a straight-line graph.

The graph of a second-degree or a quadratic polynomial is parabolic.

The graph of a third-degree or a cubic polynomial: A cubic function has a bit more variety in its shape. This graph crosses the \(x\)-axis maximum of three times.

Terms of a Polynomial: Like and Unlike Terms

In a polynomial, a term may consist of (i) only constant, (ii) only one variable, (iii) product of two or more variables, (iv) a product of both the variable(s) and the constant part. The terms may be positive or negative.
Example: \(4,\,17,\,x,\,y,\,xy,\,yz,\,5xyz,\,12xy,\, – 4,\, – 17,\, – x,\, – y,\, – xy,\, – yz,\, – 5xyz,\, – 12xy,\,……\) are terms.
1. Like term: The terms having the same algebraic factors are known as like terms.
2. Unlike term: The terms having different algebraic factors are known as the, unlike terms.
Example: In the expression \(2xy – 3x + 5xy – 4\). The terms \(2xy\) and \(5xy\) are like terms because they have the same algebraic factors \(xy\), but the terms \(2xy\) and \(-3x\) are unlike terms because they have the different algebraic factors \(xy\) and \(x\), respectively.

Solved Questions on Degree of Polynomials

Q.1. Find the degree of the polynomial \( – 2x\)?
Ans: The given polynomial has one term \( – 2x\).
The exponent of the term \( – 2x =  – 2 \times {x^1}\) is \(1\).
Thus, the degree of the polynomial is \( – 2x\) is \(1\).

Q.2.What is the degree of the polynomial \( – 8a{b^2}c\)?
Ans: Here, the degree of the constant \(-8\) is \(0\)
Power of \(a\) is \(1\), power of \(b\) is \(2\), power of \(c\) is \(1\).
So, the degree of the polynomial \( – 8a{b^2}c\) is (0 + 1 + 2 + 1 = 4).

Q.3. Write the degree of the polynomial \(1 + x + {x^2} + {x^3}\).
Ans: The given polynomial has four terms, here the first term is \(1\)the second term is \(x\). The third term is \[(x^2}\)And the fourth term is \[(x^3}\).
1. The exponent of the first term is \(1 = 0\).
2. The exponent of the second term is \(x = 1\).
3. The exponent of the third term is \({x^2} = 2\).
4. The exponent of the fourth term is \({x^3} = 3\).
Thus, the greatest exponent is \(3\). Hence, the degree of the polynomial \(1 + x + {x^2} + {x^3} = 3\).

Q.4. Find the polynomial of \(2{x^2} – 3{x^5} + 5{x^6}\)?
Ans: The given polynomial has three terms:
1. The exponent of the first term, \(2{x^2} = 2\).
2. The exponent of the second term, \(3{x^5} = 5\)
3. The exponent of the third term, \(5{x^6} = 6\).
Thus, the greatest exponent is \(6\).
Hence, the degree of the polynomial \(2{x^2} – 3{x^5} + 5{x^6} = 6\).

Q.5. Find the degree, constant and leading coefficient of polynomial expression \(4{x^3} + 2x + 3\)?
Ans: The given polynomial is \(4{x^3} + 2x + 3\).
So, the degree of the polynomial is \(3\), as the highest power of the variable of the polynomial is \(3\).
Here, the constant is \(3\).
The leading coefficient is \(4\), as the leading term is the polynomial is \(4{x^3}\).

Summary

In this article, we have discussed definition of polynomial, its degree, and its importance. We also discussed different types of polynomials like Monomial, Binomial, Trinomial, etc., based on the number of terms and then Constant polynomial, Linear polynomial, Quadratic polynomial, etc. Understanding the degree of polynomials will give a solid foundation to the students in algebra.

Frequently Asked Questions (FAQs) on Degree of Polynomials

The answers to some of the most commonly asked questions on polynomials are given below:

Q.1. What is a degree \(4\) polynomial called? Ans: The degree \(4\) polynomial is called a biquadratic polynomial.

Q.2. Explain the importance of the degree of a polynomial? Ans: From the degree of a polynomial, we understand: i. whether a polynomial is linear, quadratic, or cubic, ii. the number of zeros of the polynomial equation and iii. the nature of the graph of the polynomial equation.

Q.3. Explain what is a polynomial with a degree of \(2\) called? Ans: A polynomial having a degree of \(2\) means the highest power of variables in the polynomial terms is \(2\).

Q.4. What is a degree \(2\) polynomial? Ans: The polynomial that has the degree of \(1\) is called a linear polynomial. The highest power of variables in terms of the polynomial is \(1\).

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