Polynomials in One Variable

Introduction to polynomials in one variable
Polynomials in one variable, here we need to understand each terms very clearly. Polynomial here means, a mathematical expression consisting of  a sum/subtraction/multiplication but not division of variables , constant sand exponents.One variable means  expression consisting of one letter. Polynomial with one term is called as a "Monomial", two terms as "Binomial" and three terms as "Trinomials". Terms can be separated by addition sign or subtraction sign only but not by multiplication sign.So we can say in otherwords that polynomial is a sum of monomials . And each monomial in polynomial is called as a Term.


Examples of Polynomials with One Variable
Let us see some examples to understand better the polynomials with one variable:

1. 4x

here we have one term only with single variable x

2. 2x^3 + 4x^2 - x

Here we have three term with one variable x.

3. 2x^2

Here we have one term with variable x and the exponent as 2.

Polynomial the word poly means "many" and the word nomial means "terms". so polynomial means many terms. But we are learning here polynomials with one variable.

In polynomial we have constants , variables and exponents. we need to understand these with examples.

Constants can be any number as 3 , 2 , -20 , 3/2 etc.

Variables can be any letter as x , y , z , a , b etc

Exponents will be always positive integers such as 2x^2 , 2a3, 4y5 etc

Algebra is widely used in day to day activities watch out for my forthcoming posts on what s an algebraic expression and what is an example of an algebraic expression. I am sure they will be helpful.

Degree of a Polynomial with One Variable

The largest exponent of the variable is called the degree of a polynomial with one variable.

Let us take an example to understand better:

1. 2x4 + 4x^3 - x^2 + 8

This is a polynomial with one variable with the highest degree as 4 which is also called as the highest power or exponent of the given variable.

2. 3a^2 - 5

This  is a polynomial with one variable with degree 2.

3. x

Here the degree is 1. A variable without a power is considered to be as 1 always.

Standard Form of a Polynomial in One Variable
Standard form means writing the polynomial in one variable with the highest degree first.

Let us take an example:

3x^3 - 2x^2 +5

here the polynomial in one variable is written on that order of degree( highest degree first).

It is not necessary to write it in standard form , but it helps us in doing simplification.

Let us take few more examples:

1.Add: 3x + 4x^2+5 and 8x^2 +2x

writing it in standard form

=(4x^2+3x+5) + (8x^2+2x)

add the like terms

=4x^2+8x^2+3x+2x+5

=12x^2+5x+5

2. subtract: (3a3+ 8a^2+3) -(2a^2- 3a3+8)

write it in standard form

=(3a3+8a^2+3)-(-3a3+2a^2+8)

Reverse the sign

=3a3+8a^2+3 +3a3-2a^2-8

now add/ subtract the like terms

=3a3+3a3+8a^2-2a^2+3-8

=6a3+6a^2-5

3. Multiply: 3x^2 by 2x^3

Multiply the constant and add the exponent of the variable

=6x5

Conclusion

Polynomials in one variable is easy to simplify.

If we add / multiply polynomials in one variable we get the results as a polynomial itself.

Always while adding polynomials simply add the like terms( variable with same degree) together.

In case of multiplication , multiply the constants first then the variable and then add the degree of the variable.

While doing the subtraction change the sign to reverse sign of each term you are subtracting.