Polynomials – Study the Definition and Degree Of Polynomials

Polynomials are yet another interesting concept in the mathematics subject. Variables and Coefficient make up the polynomials. The variables are also termed as indeterminates in mathematical terminology. We perform many mathematical operations like addition, subtraction, multiplication, expansion, etc, all these are performed but not the division of variables are performed.  X2+x-12 is an example of a polynomial with a single variable but with three terms. The three terms are x2, -12, and x. 

Origin of the Word Polynomial

The word Polynomial is derived from the word ‘Poly’ which means ‘many’ in Greek. ‘Nominal’ means ‘terms’, so if we coin it together it will be read as ‘many terms’. A polynomial can have any number of definite terms.

Define the Term – Polynomial

As observed, Polynomial is the combination of two terms – Many and Terms. Thus, a polynomial is a mixture of many variables, constants, and exponents. These data are combined by using mathematical operations such as addition, subtraction, multiplication, and division. There are also many types of Polynomials depending on the number of terms present there. We will be discussing this in our upcoming section. 

Define Degree of Polynomial

The highest degree of a monomial in the polynomial term is known as the degree of polynomial. So, in order to find the degree of a polynomial, you have to look for the largest exponent in the variable. Example: 6×4+3×3+3×2+2x+1, the degree of the polynomial here is 4. 

What Are the Terms of a Polynomial?

The terms of polynomials can be identified as those elements in an equation which are generally separated by the mathematical operation such as ‘+’, ‘-’ signs. The signs that separate each part are known as terms. 

What Are the Types of Polynomials? 

Polynomials are of 3 different types and are classified based on the number of terms in them. The three types of polynomials are as follows:

  • Monomial
  • Binomial
  • Trinomial

What Are the Properties of Polynomial?

There are many properties of polynomials. These are the following properties:

  1. Divisional Algorithm 
  2. Bezout’s Theorem
  3. Remainder Theorem
  4. Factor Theorem
  5. Intermediate Value Theorem
  6. Descarte’s Rule of Sign
  7. Fundamental Theorem of Algebra

What Are Polynomial Equations? 

Polynomial Equations can be defined as those equations which consist of polynomials on either side of the equation. The polynomial equation can be arranged in a non-standard form, which we have to re-arrange in a standard form and solve the equation. The standard form will be done if we sequence the terms with the highest power to the terms with the lowest power.  

 

How Can I Solve Polynomial Equations?

The first approach to solve the polynomial equation is to think of the polynomial terms as each block. Then we need to solve each block in that polynomial equation. We will take away to differentiate each block from the other block. Below we are presenting some points so that we can easily solve the polynomial equations. The pointers are accordingly:

  • Suppose the polynomial equation is in the non-standard form then the first sequence it in the standard form. The polynomial expressions are to be placed on the left-hand side while the 0s are to be placed on the right-hand side.
  • Now simplify the polynomial expression in an equation form and then estimate the roots or zeroes in the equation.
  • If the expression is estimated to be in the form of a polynomial equation or in a quadratic equation then you have to apply the formula of the same (refer to the previous concept here).
  • If you observe that the polynomial equation has a higher degree like 3 or more than that the estimate the rational factor or the term zero.
  • Repeat the same until the residue is a linear equation or a constant.   
  • Now you have to list all the zeroes or the roots which you have. 

 

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