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.

Introduction of add polynomials

Introduction of add polynomials:

          A polynomial is an expression formed by variables and constant, using the  fundamental operation addition, subtraction, multiplication, and non-negative, number exponents. For example, x 2 - 4x + 5 is a polynomial. An algebraic expression, ax 2 is called a monomial in x, x is a variable a. The number a is called the coefficient of x^2 , the degree of the monomial. The sum of two monomials is called a binomial. The sum of a limited number of monomials in x is called a polynomial in x.

Please express your views of this topic solve a math problem by commenting on blog.

Polynomials and Methods for Adding

The monomials in a polynomial are called the terms and the highest degree of the terms, the degree of the polynomial. The coefficient of the highest power of x in a polynomial is called the leading coefficient of the polynomial. For example, 3x – 1 + 5x^2 + 6x^5 – 4x3 is a polynomial of degree 5 with terms 2x, –1, 5x^2, 6x^5 and –4x3 and most important coefficient 6. The terms of a polynomial are real numbers, they are rearranged using the property of real numbers

A first degree polynomial is called a linear polynomial. A second degree polynomial is called a quadratic polynomial. A degree polynomial is third called as cubic polynomial and a degree polynomial is four called as bi-quadratic polynomial. Since the terms of a polynomial are numbers, all polynomials represent numbers. So we can add, subtract, multiply and divide polynomials. The following method used to add the polynomials.

Adding Polynomials Examples

Ex 1: Find the sum or add of 3x^4 – 4x^2 + 5x + 3 and 4x + 6x3 – 6x^2 – 2 polynomials.

Sol :  Using the properties of real numbers, we add the polynomial

(3x^4 – 4x^2 + 5x + 3) + (6x3 – 6x^2 + 4x – 2) = 3x^4 + 6x3 – 4x^2 – 6x^2 + 5x + 4x + 3 – 2

= 3x^4 + 6x3 – (4+6)x^2 + (5+4)x + 1

= 3x^4 + 6x3 – 10x^2 + 9x + 1.

The following method is useful in add two polynomials

3x^4 + 0x3 – 4x^2 + 5x + 3

0x^4 + 6x3 – 6x^2 + 4x – 2

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3x^4 + 6x3– 10x^2 + 9x + 1

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Polynomials

We have worked on for monomials in my previous post, today lets work on the introduction of Polynomials. We will learn on how to solve polynomials in my next post. This is a informative topic, and i am sure you will enjoy working on it as well.

Introduction of Polynomials:-

Its an algebraic expression which the variables have only non negative powers using the operations of addition, subtraction, multiplication.

Addition and subtraction of two polynomials mean like terms. Multiplication and division also can be performed.

There are various methods of factorizing a polynomial. They are,

1. Factorization by dividing the expression by the HCF of the terms of

the given expression.

2. Factorization by grouping the terms of the expression.

3. Factorization using identities.

Example of Polynomials are as mentioned below.

P(x) =9x⁵ + 7x⁴ + 2x³ + 6x² + 6x + 1

G(x) =19x + 45

I will work more on this in my next post, watch for more information on polynomials. Give me your feedback on this post as well.

Also you can follow my blogs for Precalculus Problems. I hope this was useful.