What is a Polynomial? Definition and Types

An algebraic expression in which the variables involved are free of having negative integer powers is called a polynomial. A polynomial consists of variables, constants and coefficients.

Examples: x²+2y, x⁴-7, etc.

Below given are some algebraic expressions. Let us try to identify the polynomials in them.

x³, x^1/2-2y, 5y²+2/z, 3x²-xy+4y³

If you observe carefully,x³, 3x²-xy+4y³ are the only polynomials given in the above algebraic expressions.

Whereas x^1/2-2y, 5y²+2/z are not polynomials. As we already discussed, only variables of non – negative powers are polynomials. Hence, 5y²+2/z can be written as 5y²+2z^-1 which has a negative exponent, i.e., (−1), and x^1/2-2y can be written as √x-2y, which has a fractional exponent.

Note that variables containing radicals (square roots), algebraic expressions with fractions, and having variables in the denominators are not polynomials.

POLYNOMIALS IN ONE VARIABLE

Variables are denoted by symbols that can take any value. Let x, y, z, etc., be denoted as variables. So, we have the algebraic expressions like 5x, -7x,5/8x,…………… all polynomials in one variable. These are of the form, a constant multiplied by some power of a variable. You are also familiar with algebraic expressions like x³+4x-1, x⁴-3x+1. All these expressions are polynomials in one variable. For example, If x = 2 for the above expression 5x, then 5(2) = 10.

Note: If the algebraic expression containing two variables like 2x-3y,6x²+5y³+1,….. are all polynomials in two variables.

Let’s look at the below table describing types of polynomials according to their number of terms.

DEGREE OF THE POLYNOMIAL

Let us consider a polynomial a(x), where x is the variable, and a is constant. Now, the highest power of x or the highest exponent of x in polynomial a(x) is called the degree of the polynomial.

Example: In the polynomial 8x – 1, the power of x is 1. Therefore, it is a polynomial of degree 1.

Let’s find out the degree of given polynomials:

(i)4x³-y-1     (ii)5xy⁴+2xy²-6

The degree of (i) is 3 since the highest power of the variable is 3, and the degree of (ii) is 4 since the highest power of the variable is 4.

Simply we can say that the degree of the polynomial is the largest degree of its variable term.

A polynomial in one variable z of degree n can be written in the form,

b_nzⁿ+b_n-1z^n-1+……+b₁z+b₀

Where b₀,b₁,…….., b_n are constants and b_n≠0.

In the above expression, if all the constants (coefficients) are zero, i.e., b₀=b₁=b₂=…b_n=0, then the polynomial is called a zero polynomial.

Let's look at below table of types of polynomials according to a degree:

ZEROS OF A POLYNOMIAL

Let us consider a polynomial px=2x²+2x-12

Substitute x = 2 in the polynomial, then

p(2)=2(2)²+2(2)-12

=8+4-12

=12-12

=0

Hence, p(2) = 0 is called the zero of the polynomial.

Simply, we can say that zero of a polynomial p(x) is the value of x, for which p(x) = 0.

PROPERTIES OF POLYNOMIALS

Dividend = Divisor × Quotient + Remainder

Remainder theorem: Let p(x) be any polynomial of degree one or greater than one, and let c be any real number. If p (x) is divided by the linear polynomial (x – c), then the remainder is p(c).

Factor theorem: If p(x) is a polynomial of degree n ≥ 1 and c maybe any real number, then

1. (x – c) will be the factor of p (x), if p (c) = 0
2. And its converse, i.e., if (x – c) is a factor of p(x), then p (c) = 0.