Polynomials in one variable are algebraic expressions that are of the form ax^{n} where n is a nonnegative (i.e. positive or zero) integer and a is a real number, called the coefficient of the term. The degree of a polynomial in one variable is the largest exponent in the polynomial.
Polynomials in one variable are those expressions in which there is only one variable present. In mathematics, a polynomial is an expression that consists of variables and coefficients, involving the operations of addition, subtraction, multiplication, and exponentiation of variables. The word "polynomial" contains two words, namely, “poly" and “nomials”. "Poly" means many, and "nomials" means terms. Hence an expression containing many terms is called polynomials, having variables and coefficients.
Some examples of polynomials in one variable are given below:
To learn about the classification of one variable polynomial based on degree, let us first understand the meaning of the degree of a polynomial. The highest power of the variable in a polynomial is called the degree of the polynomial. For example, in the following equation: x^{2} + 2 x + 4, the degree of the polynomial is 2, i.e., the highest power of the variable in the polynomial.
Based on the degree of a polynomial in one variable, it can be classified into 4 types:
For a multivariable polynomial, the degree is the highest sum of powers of different variables in any of the terms in the expression.
Solving polynomials refers to finding their roots. A root or zero of a polynomial is the value for which the given polynomial is equal to zero. The first step in solving any polynomial is to find its degree. As discussed above the degree of a polynomial with one variable is the largest exponent of that variable. Thus,
Example: Find the perimeter of the square with side length 8x units. Also, state the type of polynomial obtained (based on the degree).
Solution: We know that the perimeter of a square is the sum of all its sides. In a square, all sides are equal. So, if we add all four sides of the square, we get, 8x + 8x + 8x + 8x, which equals 32x.
∴ The perimeter of the square is 32x units.
32x has only one variable with degree 1. So, it is a linear polynomial in one variable.
∴ The perimeter of the square is 32x units which is a linear polynomial in one variable.
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