This article guides students how to calculate or find out the factor of polynomials. There are some basic concepts that need to be covered while coming to the topic. Without learning this a student cannot factorize or find out the factor of any polynomial. Here, are some basic concepts of the Chapter on Polynomials.

**What is a Polynomial?**

A polynomial is nothing but an expression of variables with exponents and constant real numbers.

For example: 4x^{2} – 3x + 7

In the above example +4,-3,+7 are real numbers that are constants and x is a variable of the polynomial given in the example with exponential values 2 and 1.

However, exponents must be positive. In case of the example: 7x^{2}+ 3 , it is not a polynomial as the exponent of x is not positive. Similarly, in, the exponent of x is in a negative integer.

Degree of a Polynomial:

The degree of a polynomial is the exponent with the highest value. For example: 3x^{5}-2x^{2}+x-10

Here, we can see that the polynomial consists of four terms 3x^{5}, -2x^{2}, x, and -10, with the variable having different exponents x^{5}, x^{2} x^{1} and x^{0} . The highest exponential value in this polynomial is in the variable x with exponent 5. Hence, the degree of the polynomial in variable x is 5.

Based on the degree of a polynomial, there are various naming for different polynomials.

**A Constant Polynomial:** A constant polynomial is one with the degree of polynomial equal to 0 (zero)

For example: f(y)= 3 is a constant polynomial as 3= 3 X y^{0}=3×1=3 which is a constant value.

**Linear Polynomial:** A linear polynomial has degree one (1).

For example: f(y)=y+7, h(x)=-3x+6, etc. are linear polynomials because their degree of polynomial is 1.

**Note*** – A Linear polynomial can either be monomial i.e., consisting of one term, binomial consisting of two terms

Example: f(y)=7x is monomial whereas g(x)=2x+7 is binomial.

**Quadratic Polynomial:** A polynomial with the degree of polynomial equal to two (2) is called a quadratic polynomial.

Example: 9y²-5y-3, with 2 as the degree of polynomial

**Note***– It can be either monomial or binomial, or trinomial .

**Cubic Polynomial:**A polynomial with degree 3 is called cubic polynomial.

For example: f(x)= 3x³+2, is a cubic polynomial, with degree 3.

After knowing what polynomials are, different types of polynomials, related terms and the degree of polynomials, we will now see how we can find the root and value of any polynomials.

- The
**value**of any polynomial let’s say g(x) at constant β is achieved by substituting x=β in the given polynomial, i.e., f(β).

Example:

To find the value of the polynomial x²+6x-5 at x=1 and x=2

- We first substitute x=1 in polynomial

i.e., 1²+6(1)-5

the value of the polynomial =1+6-5=2

- Next the we substitute x=2,

i.e., 2²+6(2)-5

the value of the polynomial= 4+12-5=11

Also, the value of the polynomial y²+2y-24 at y=4 is

=4²+2(4)-24

=16+8–24

=24–24

=0

Here, 0 is the value of the polynomial is at y=4 which shows that y=4 is the root of given polynomial.

**Zero or Root of Polynomial:**This polynomial is obtained when the value of the polynomial is 0 i.e., g(x) is zero at g(α) when we substitute y=α.

Example: Find out whether y=2 is a root of polynomial g(y)=y²+5y-9?

Solution:

g(2)=2²+5(2)-9

g(2)=4+10–9

g(2)=5

Therefore, the value of g(x) is 5 which is ≠0 at x=3. This shows that y=2 is not a zero or root of the polynomial.

**Rational Root Theorem**: This theorem states that if is the root of the polynomial g(x) in which the leading coefficient does not equate to 0, then we can say that the leading term has a factor c and the constant term with factor b.

Example: Find the root of a polynomial g(a)=2a²+6a-8,

In this example, the leading coefficient is ≠ 0,

Therefore, b is factor of content term 8, which include ±1, ±2, ±4, ±8 and the leading coefficient 4 has factor c = ±1, ±2.

Therefore, the possible root of polynomial g(a) is ±1, ±2, ±4, ±8, ±

Now by trial and error method, one can verify all the possible value of g(a) by checking g(a) at a=1,2,4 and so on to find the root of polynomial g(a).