7th Class Mathematics Algebraic Expressions Algebraic Expression

     Introduction

We have discussed about the addition, subtraction, multiplication and division of the arithmetic expression into previous chapter. In this chapter, we will discuss about the operation on algebraic expression.  

     Algebraic Expression

It is the combination of constants and variables along with the fundamental operations \[(+,\,-,\,\,\times ,\,\,\div )\]

Terms: It is the part of an algebraic expression which is separated by the sign of addition and subtraction.

\[5{{x}^{4}}{{y}^{2}},35{{x}^{4}}{{y}^{2}}-13{{x}^{2}}y,6xy,-3\] is an algebraic expression having \[8{{x}^{3}}{{y}^{2}},-4{{x}^{2}}y,6xy,-3\] as its term.                

      Like and Unlike Terms

The terms having similar variable(s) are called like terms otherwise it is unlike. In an algebraic expression \[5{{x}^{4}}{{y}^{2}},-13{{x}^{2}}y+6xy-3-35{{x}^{4}}{{y}^{2}};5{{x}^{4}}{{y}^{2}},35{{x}^{4}}{{y}^{2}}\]are like terms and are unlike terms.  

      Types of Algebraic Expression  

Monomials

An algebraic expression which contains one term is called monomial. i.e. \[3x,4x,-xy,{{b}^{2}}a\] etc. are monomials.  

Binomials

An algebraic expression which contains two terms is called binomial. i.e. \[x+y,a-b,{{b}^{2}}a+{{a}^{2}}b\] etc. are binomials.

Trinomials

An algebraic expression which contains three terms is called trinomial.

i.e. \[\left( x+y+z \right),\text{ }\left( a+b+c \right),\text{ }\left( {{a}^{2}}+{{b}^{2}}+{{a}^{2}}{{b}^{2}} \right)\]etc. are trinomials.

i.e. \[\left( a-b-c+d \right),\left( x+y+z-c \right)\]etc. are quadrimonials.  

      Polynomials                

An algebraic expression in which the variables have only non-negative integral power is called polynomial.

• \[5{{a}^{3}}-4{{a}^{2}}+6a-3\]is a polynomial because powers of variable "a" are non-negative.
• \[7{{a}^{3}}{{b}^{2}}-4{{a}^{2}}b+6ab-3\] is a polynomial in two variable.
• \[4{{x}^{2}}+23{{x}^{3}}+37xy+45\]is a polynomial in two variable.
• \[\sqrt{2x}+3{{x}^{2}}+5\]is not a polynomial because the power of first term is \[\frac{1}{2}\]which is not a non-negative integer.
• \[4{{e}^{2}}+\frac{1}{6}e+2\sqrt{3}\]is a polynomial in one variable.

      Degree of Polynomial

In case of one variable, the highest power of variable is the degree of polynomial e.g \[5{{x}^{4}}-4{{x}^{2}}+6x-3\] is a polynomial of degree 4. If polynomial is in the more than one variable then the highest sum of degree of the variables in each term is the degree of polynomial. e.g. \[8{{x}^{3}}{{y}^{2}}-4{{x}^{2}}y+6xy-3\]is a polynomial of degree 5.  

      Linear Polynomial

A polynomial of one degree is called linear polynomial.

\[p\left( x \right)=6x-3\] is a linear polynomial.  

      Quadratic Polynomial

A polynomial of two degree is called quadratic polynomial.

\[q(x)=4x2+3x+45\] is a quadratic polynomial.