Degree of a Polynomial

Introduction

The Degree of a Polynomial is the largest of the degrees of the individual terms. Polynomials are a type of algebraic expression in which the exponents of all variables must be the whole number. The exponent of the variable in each polynomial must be a non-negative integer. Polynomials can be thought of as dialects of mathematics. They are used to represent numbers in almost every area of mathematics and are used in specific areas of mathematics.

You can perform arithmetic operations such as polynomial additions, polynomial subtractions, polynomial multiplications, and exponentials of positive integers, but you cannot perform variable divisions.

In this tutorial, we will discuss polynomials and the degree of polynomials.

Polynomials

Polynomials are a type of algebraic expression in which the exponents of all variables must be the whole number. The exponent of the variable in each polynomial must be a non-negative integer. Polynomials are made up of constants and variables, You can perform arithmetic operations such as polynomial additions, polynomial subtractions, polynomial multiplications, and exponentials of positive integers, but you cannot perform variable divisions.

For example, 6x²+ 5x +5 is a quadratic polynomial.

Terms of a polynomial

Polynomial terms are generally part of an equation separated by a "+" or "-" sign. The "term" of a polynomial is the product of a number and one or more variables, which is the power of a positive integer. That is, there is no root or variable division.

Finding a value (or values) of the variable that satisfies the equation is the first step in solving a quadratic or cubic equation. The value(s) that the quadratic or cubic equation requires is referred to as its roots, solutions, or zeros.

Degree of a Polynomial

The degree of a monomial is the sum of the powers of the exponents of its variables. The degree of a polynomial is the highest degree of a monomial (single term) with a non-zero coefficient. The degree of the term is a non-negative integer.

For example, P(x)=6x³+ 2x²+1

The order of this cubic equation is 3.

Constants

A fixed value that appears in algebraic expressions and equations is known as a constant. A constant has a fixed value and does not change over time.

x+y = 8 is an algebraic expression, and 8 is a constant that cannot be altered.

Variables

The words that vary or vary throughout time are known as variables. Unlike constant, it does not stay the same. As an illustration, a person's height and weight are variables because they are not always constant.

x and y, the variables in the algebraic formula x+y = 8, are interchangeable.

Different Polynomials on the basis of the degree

The degree of the polynomial is defined as the power of the leading term or the highest power of the variable. This is accomplished by arranging the polynomial terms in descending order of power. They are classified into four major types based on the degree of the polynomial.

• 0 or constant polynomial − Polynomials with zero degrees are known as zero polynomials.

• Linear Polynomial − Polynomials with a degree of one are known as linear polynomials. In a linear polynomial, the highest exponent of the variable(s) is 1

• Quadratic Polynomial − Quadratic polynomials are two-degree polynomials.

• Cubic Polynomial − Cubic polynomials are polynomials of three degrees.

Relation between coefficients and zeros for the degree 2 and 3

There is a clear mathematical connection between the zeroes and the coefficients, and the number of zeros in a polynomial equals the degree of the polynomial.

For quadratic equations: Let the equation is ax² + bx + c = 0, and their roots are α and β

Then, The Sum of roots = $\mathrm{α +β=\frac{-b}{a}}$

And, Product of roots = $\mathrm{α ×β=\frac{c}{a}}$

For cubic equations: Let equation is ax³+bx² + cx + d = 0 and their roots are α, β and Υ

Then, Sum of roots = $\mathrm{α +β+γ=\frac{-b}{a}}$

And, Product of roots = $\mathrm{α ×β×γ=\frac{-d}{a}}$

and $\mathrm{αβ+βγ+γα=\frac{c}{a}}$

Solved Examples

1) Find the zero of linear polynomial 8x – 8.

Ans: Given linear polynomial is 8x – 8 to find its zero, first equate it to zero

$$\mathrm{8x – 8=0}$$

$$\mathrm{\Rightarrow 8x=8}$$

$$\mathrm{\Rightarrow x=1}$$

Hence, the zero of the given polynomial is 1.

2) Factorise the given polynomial x²+3x+ 2

Ans: Given a quadratic polynomial x²+3 x+2

$$\mathrm{x^2+3 x+2 = x^2+2 x+x+2}$$

$$\mathrm{=x ( x+2 )+( x+2 )}$$

$$\mathrm{=( x+2 ) ( x+1 )}$$

3) Find the zero of the quadratic polynomial x²+4x+4

Ans: Given a quadratic polynomial x²+4x+4 for finding its zero first equate it to 0.

$$\mathrm{x^2+4x+4= (x+2)^2=0 }$$

$$\mathrm{\Rightarrow x=-2 }$$

Hence the zero of the given polynomial is -2

4) Simplify the given polynomial ( x²+2 x+1 )×( x+1 )

Ans: The given polynomial is (x²+2x+1)×(x+1)

$$\mathrm{\Rightarrow (x^2.x+2x.x+1.x)+(x^2+2x+1)}$$

$$\mathrm{\Rightarrow (x^3+2x^2+x)+(x^2+2x+1)}$$

$$\mathrm{\Rightarrow x^3+3x^2+3x+1}$$

5) Simplify (x+4)²=2 by using identities.

Ans: Given the equation is (x+4)²=16

$$\mathrm{x^2+4^2+8x=16}$$

$$\mathrm{\Rightarrow x^2+8x=0}$$

$$\mathrm{\Rightarrow x(x+8)=0}$$

$$\mathrm{\Rightarrow x=0\: and\: x=-8}$$

6) Simplify (x-2)²=1 by using identities.

Ans: Given equation is (x-2)²=1

$$\mathrm{x^2+2^2-4x=1}$$

$$\mathrm{\Rightarrow x^2-4x+3=0}$$

$$\mathrm{\Rightarrow x^2-3x-x+3=0}$$

$$\mathrm{\Rightarrow x(x-3)-1(x-3)=0}$$

$$\mathrm{\Rightarrow (x-3)(x-1)=0}$$

$$\mathrm{\Rightarrow x=1 \: and\: x=3}$$

Conclusion

Polynomials are a type of algebraic expression in which the exponents of all variables must be the whole number. A polynomial can have any (finite) number of terms. The general expression will be given as

$$\mathrm{f(x)=a_n x^n +a_{ n - 1} x^{n-1}+ a_{n-2} x ^{n -2} +................+ a_1 x^{1} + a_0}$$

where a₀,a₁,a₂,a₃,a₄,........ are the coefficient of the polynomial and x is a variable whose power must be only the whole number.

FAQs

1. What do you mean by a polynomial?

Polynomials are an algebraic expression type in which the exponents of all variables must be the whole number. A polynomial can have any (finite) number of terms.

2. What is the general expression for polynomials?

The general expression will be given as

$$\mathrm{f(x)=a_n x^n +a_{ n - 1} x^{n-1}+ a_{n-2} x ^{n -2} +................+ a_1 x^{1} + a_0}$$

where a₀,a₁,a₂,a₃,a₄,........ are the coefficient, and x is a variable whose power must be only the whole number.

3. What do you mean by the degree of a given polynomial?

The degree is the sum of the exponent of a monomial (single term) with the highest degree. The degree of the term is a non-negative integer because it will be the sum of the exponents of the variables contained.

4. What is the general equation for quadratic equations?

The general equation for the quadratic equation is ax₂+ bx + c = 0.

5. What are the zeroes of a polynomial?

The zeros of a given polynomial are the points where polynomials are zero. The zeros of a polynomial is a point for which the polynomial becomes zero. On putting that value in the polynomial, the value of the whole polynomial becomes equal to zero.