Importance of Polynomials

In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number.


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Importance of Types of polynomials:

It is necessary to know the importance of types of polynomials. The importance of the types of polynomials are as follows:

(1) Constant polynomial

(2) Linear polynomial.

A polynomial where its degree is zero is called as a zero polynomial.

For example, f(x) = 7, g(x) = 7/4 are constant polynomials.

A polynomial where its degree is 1 is called as the linear polynomial.

For example, p(x) = 4x-3, q(y) = 3y are the linear polynomials.

Both the sum and the product of a polynomial is a polynomial and the derivative of a polynomial function is also a polynomial function.

Importance of concepts of polynomials with examples:

The followings are some of the important concepts if polynomial.

If p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial.

A polynomial of degree one is called a linear polynomial.

If a polynomial of degree is two then it is called a quadratic polynomial.

If a polynomial of degree is three then it is called a cubic polynomial. It has a general form of ax3 + bx2 + cx + d.

Each term of a polynomial has a coefficient in the form of –x3+ 4x2+7x-2. The coefficient of x3 is -1, the coefficient of x2 is 4.

Examples:

1. Check whether -2 and 2 are zeros of the polynomial x + 2.

Solution:

Let p(x) = x + 2

Then, p(2) = 2 + 2

= 4

p(-2) = -2 + 2

= 0

Therefore, -2 is a zero of the polynomial x + 2, but 2 is not.

2. Determine whether (x–3) is a factor of the polynomial p(x) = x3 – 3x2 + 4x – 12.

Solution:

For (x–3) to be a factor of p(x),

p(3) should be zero by the factor theorem.

Now p(3) = 33 – 3(3)2 + 4(3) – 12

= 27 – 27 + 12 – 12

= 0

Hence (x–3) is a factor of the given polynomial.