Basic Concepts of Polynomials and their utilization

A Polynomial is defined as an expression that consists of more than two or multiple algebraic terms. Polynomials are majorly the sum of many terms with their powers or exponents of unknown variables. There are a few things related to polynomials. Adding or subtracting a polynomial gives another polynomial. Similarly, multiplying a polynomial with another polynomial results in a different polynomial. Polynomial is a Greek word where poly means multiple and Nominal means terms. Thus the word Polynomial means multiple terms in English. Polynomials represent a function. So if we graph it, there would be a smooth and curvy line without any breaks.

What does a polynomial contain?

A polynomial mainly consists of variables, exponents, coefficients, constants, and operator symbols. For example, 9×2 + 3y +7 is a polynomial expression.

  1. Variables: In the above example, x and y are the variables.
  2. Exponents: Here exponents are denoted by the powers of the variables. In the above example, power 2 denoted with the variable x is called the exponent.
  3. Coefficients: They are always attached to the variables. From the above example, we can say that numbers 9 and 3 written before the variables x and y are called coefficients.
  4. Constants: Constants are numbers found within a polynomial expression. Here, by looking at the above example, 7 is known as the constant within the expression.
  5. Operators: Operators are usually denoted by addition, subtraction, multiplication, and division symbols. In the above example, we are using multiplication and addition operators which can be formed as 9 * x2 + 3 * y + 7.

Few rules of Polynomial Expressions

  • Polynomials cannot be divisible by a variable. The expression 3×2 + 2y/4 is a polynomial since it is divisible by 4 which is not a variable. But the expression 3×2 + 2y/y + 1 is not a polynomial since it is divisible by the variable, y +1.
  • Polynomials do not contain negative powers or exponents. For example, 3x-2 + 2y-4 is not a polynomial expression, since it contains negative exponents. If we try to convert it into a positive one, we need to use a division operator which cannot be used in polynomials. x-2 can be written as 1/x2 and we know that polynomials cannot be divisible by a variable.
  • Polynomials cannot have fractional powers or exponents. For example, 3×2 + 2y1/2 + 5 is not a polynomial since it is using fractional exponents.
  • Polynomials cannot have radicals within the expression. Within the expression, 2×2 + √3y + 2 is not a polynomial since it contains the radical symbol √.

What is the degree of a polynomial?

To find the degree of polynomial, we have to jot down the terms of the polynomial in decreasing order. The terms containing the highest power of the exponents are considered the leading term and represent the degree of a polynomial. Usually, the degree of the polynomial is represented by adding up the exponents.

For example, let us find out the degree of a polynomial from the expression, 4x3y4 + 6x2y + 7x. So let us add the exponents for the terms.

Thus, for the first term, 4x3y4, the exponents can be denoted as 3 from x3 and 4 from y4. Thus, adding them up we would get 7. For the second term, 6x2y, the exponents can be denoted as 2 from x2 and 1 from y. Thus, the total comes up to 3. Now, the last term 7x consists of only 1 exponent with a degree of 1. Hence, in the first term, the highest degree is denoted by 7 and is the leading term. So we can conclude that 7 is the degree of the polynomial.

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