Important algebra terms || Basics of algebra

In this post we will try to understand various terms used in algebra with examples.

These terms will help to understand the concept of algebra better and allow you to solve questions faster.

Important algebra terms with examples

Algebraic expression

The whole collection of constants and variables (or both) separated by addition or subtraction sign is known as algebraic expression.


Examples of algebraic expression

(a) 2 + y

Here the constant “2” and variable “y” is separated by addition sign “+”.

(b) 2x – 3xy

Here the expression involves two entities:
2x ⟹ made of constant “2” and variable “x”
-3xy ⟹ made of constant “-3” and variable “xy”

(c) x + y + z

The expression contain three entities.
Variable x , y and z separated by addition sign ” + “

Note:
Addition / Subtraction is used to separate the given entities.
Constants or variables with multiplication/division sign is one entity only.

(d) x.y.z

The expression contains one entity.
Here the variables; x , y & z are multiplied together.

Terms of expression

The entities in the expression separated by addition/subtraction sign is called terms.

The algebraic expression may consists of many terms of different constants and variables.

Examples of Terms

(a) 7x + 3y + 2

Here the algebraic expression consists of three terms
7x ⟹ made of constant “7” and variable “x”
3y ⟹ made of constant “3” and variable “y”
2 ⟹ constant 2



(b) \mathtt{7x^{2} y +\ 3xy\ +\ 2x}

The algebraic expression is made of three different terms.

\mathtt{7x^{2} y}
⟹ 3xy
⟹ 2x

(c) 9x – 4

The expression consists of two terms 9x and -4

Variable in Algebra

A variable is a term in algebraic expression which doesn’t have any fixed value.

Generally, the variable is expressed in form of alphabet.

Example
In expression 2 + 3y, the term “y” is a variable.

Similarly in expression y + 2xy, the term “x” and “y” is a variable.

Constant in Algebraic expression


The term whose value is fixed and do not change is a constant.

All the numbers in math are the form of constants.

For example;
In the expression 2x + 7, there are two entities.

2x ⟹ made of constant “2” and variable “x”
7 ⟹made of constant “7”

Coefficient in algebraic expression

The constant number multiplied with the variable is called coefficient.

The coefficient is always a constant number.

For Example;
Consider the entity 2x.

Here the constant number 2 is multiplied with variable x.
Hence, number 2 is the coefficient.


Example 02
6xy + 7y

Here the expression consists of two entities.

6xy ⟹ coefficient is 6
7y ⟹coefficient is 7

Algebraic equation

It is a collection of terms with an equal sign.

The equation tells that the left side of the terms is equal to the terms on right side.

For example;

(a) x + 5 = 2

Terms on left side ⟹ x + 5
Terms on right side ⟹ 2

Its a math equation which tells that for given value of x; terms on left side is equal to terms on right side.

Can we find the exact value of x for the equation?

Sure, just simplify the given equation.

x + 5 = 2

Taking 5 on the right side.

x = 2 – 5

x = -3

Hence for x = -3, the above equation satisfied.

Validation
Put x = -3 in the equation

-3 + 5 = 2

2 = 2

Hence, when we put x = -3, the left side becomes equal to right side.

Example 02
2x + 9 = 19

It’s an example of equation.
The equation tells that for particular value of x, the left side becomes equal to right side.

Let’s find the value of x by simplifying the equation.
2x + 9 = 19

Take 9 on right side.

2x = 19 – 9

2x = 10

Divide both side of equation by 2, we get;

x = 10/2

x = 5

Hence for x =5, both side of the equation become equal.

Exponents of Variable

The exponent of variable tells the number of times the variable is used in the entity.

The exponent is expressed in the form of power above the variable.

For example:
\mathtt{\ x{^{3}} +\ 4x^{2} \ +\ 2x}

There are three entities in the expression.

\mathtt{\ x{^{3}}} ⟹ exponent of x is 3

The variable x can also be expressed as x. x .x


\mathtt{4x^{2}} ⟹ exponent of x is 2 It means that x is multiplied two times, (i.e. x . x)</p> <p style="font-size:18px">2x ⟹ exponent of x is 1.</p> <p style="font-size:18px"> </p> <p style="font-size:18px"><strong>Why exponent is used?</strong> Because writing any variable multiple times gets very complicated.</p> <p style="font-size:18px">Let's suppose there is a variable y appearing 5 times.</p> <p style="font-size:18px">This can be written as : y . y . y . y . y</p> <p style="font-size:18px">This looks very complicated.</p> <p style="font-size:18px">To avoid writing multiple y's we can use exponent to write: [latex] \mathtt{y^{5}}

Example 02
Write y . y . y . z . z in the form of exponents

There are three y's and two z's multiplied together.

We can write exponents as : [/latex] \mathtt{y^{3} .\ z^{2}} [/latex]

What are polynomials?

Polynomials are algebraic expression containing constants, variables and exponents.

Now there are different types of polynomial:

(a) Polynomial with one entity is called monomial

Example:
⟹ 4y
\mathtt{9\ y^{2} \ z}
\mathtt{10xy}



(b) Polynomial with two entities is called Binomial

Example:
⟹ x + y
⟹ 2 + 9x
\mathtt{3x+\ y^{2} \ z}


(c) Polynomial with three terms is called Trinomial

Example:
⟹ x + y + z
⟹ 2 + x + 3y
⟹ xy + yz + zx

Degree of Polynomial

The highest number of variable present in the entity is called degree of polynomial.

For example;
\mathtt{x\ +\ xy\ +\ x^{3} y}

There are three entity in the polynomial
x ⟹ one variable
xy ⟹ two variable
\mathtt{x^{3} y} ⟹ Four variable

Highest number of variable in the polynomial is 4.
Hence, the degree of polynomial is 4

Example 02
\mathtt{1+xy^{2} z^{2} +x^{2} y+xyz}

There are 4 entities in the polynomial
1 ⟹ 0 variable
\mathtt{xy^{2} z^{2}} ⟹ 5 variables
\mathtt{x^{2} y} ⟹ 3 variables
xyz ⟹ 3 variables

There are maximum of 5 variable in the entity.
Hence the degree of polynomial is 5








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