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Standard Algebraic Formats

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Last updated date: 17th Apr 2024
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Algebraic Formats: Introduction

The standard algebraic format deals with the symbolic presentation of any valid mathematical statement in terms of some specific algebraic expressions, identities, or equations. Here, the different combinations of mathematical operations like division, multiplication, addition, subtraction, and exponentiation are executed about the conditions of the problems. The mathematical operators follow a specific " BODMAS " sequence to arrive at the correct answer to any algebraic expression.


What are the Parameters Associated with an Algebraic Format?

The following parameters associated with an algebraic format may be present in algebraic expressions, identities, or equations.

  • Constants: A quantity that has a defined value and does not change throughout the operations done in the algebraic format. Examples are the numerical values, or any letters assumed to be constants.

  • Variables: A variable is the symbolic representation of an unknown value. We usually represent the variables using letters such as x, y, or t. For instance, when we declare that l stands for the length of a rectangle and w stands for the width of the rectangle; we can represent its area A as $A=l\times w$.

  • Coefficients: The coefficient is the constant that accompanies the variable.
    For example, in the entity $3 \times y$ or$3y$, 3 is the coefficient of the variable y.

  • Terms: All the individual entities of an algebraic expression, identity, or equation, which are separated by the operators, are commonly known as the terms. Thus, a term may be a constant or a product of a coefficient and a variable.


In the algebraic equation $x+3=3{{y}^{2}}+2z$, there are four terms, namely, $x$, $3$, $3{{y}^{2}}$, $2z$.


What are Algebraic Expressions?

The mathematical expression containing one or more “algebraic variables” is called an algebraic expression. It may be along with the associated “coefficients” and “constants”. Examples of algebraic expressions are $a{{x}^{2}}+b$, ${{y}^{3}}+3$,


When the “equal to sign” symbolised as “=” is used to unite any two or more algebraic expressions, the resulting mathematical entity is called an “algebraic equation”. An algebraic equation may correlate an algebraic expression to any constant only. It suggests that the terms or expressions on either side of the “=” are equivalent.


Examples of algebraic equations are $a{{x}^{2}}+b=c$, $ {{y}^{3}}=3$.


Terms of Equation


Terms of Equation


What are the Types of the Algebraic Equations?

According to the number of terms used to form an algebraic expression, some of the algebraic equations are explained below.


  • Monomial: An algebraic expression having just one term is called a monomial. for instance, etc. $3y,2xyz,4x,-xy,\dfrac{{-5}}{3}abc,$, are monomials.

  • Binomial: An algebraic expression that contains two unlike terms is named a binomial. for instance , $x + y,4p + 2z,3{x^2}-{y^2}$etc., are binomials.

  • Trinomial: An algebraic expression having three unlike terms is named a trinomial. for instance , $a + 4b + 2z,x-pq + yz,$etc., are trinomials.

  • Quadrinomial: An algebraic expression containing four terms is named a quadrinomial. for instance , $7ab-c + z + 4xy,abc-a-b-c$ are quadrinomials.

  • Multinomial: An algebraic expression with two or quite two terms is called a multinomial. for instance, $4x + 3,5-x,{y^2} + 7y$, each may be a multinomial of two terms is a multinomial of four terms. $-ab + 7b-6bc-z,a-ab + 7b-6bc-z$a may be a multinomial of five terms, and so on.

  • Polynomial: An algebraic expression with one or more unlike terms with the facility of the variables as only whole numbers is called a polynomial. In other words, all monomials, binomials, trinomials, and every other expression having any number of finite terms with the power of their variables as whole numbers are called polynomials. No term during a polynomial contains a negative exponent or any variable in the denominator. for instance $4{{x}^{3}}+3{{x}^{2}}+9x-1,9{{a}^{2}}+5,4{{x}^{2}}+3{{x}^{2}}+9x-1,9{{a}^{2}}+5$, are polynomials.


What are Algebraic Identities?

The identities dealing with only the algebraic expressions that may contain coefficients of algebraic variables and constants are known as “algebraic identities”.


Algebraic identities are always true for any value of the algebraic variables involved in one or both the expressions on either side of the “=”.


The mathematical identity ${{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ is an example of an algebraic identity.


What are the Advantages of Algebraic Identities?

The following are some of the advantages of algebraic identities.


  • An algebraic identity does not show a finite number of solutions, as it is true for an infinite number of values you assign to its variables.

  • The algebraic identities are useful in the simplification of algebraic polynomials.

  • The algebraic identities may be utilised in the factorisation of algebraic expression

  • The algebraic identities also help solve any algebraic equation.


What are Algebraic Polynomials?

Algebraic polynomials are expressions or equations possessing only the non-negative integral power of variables of the algebraic expressions or equations but may contain any integral coefficients and constants.


Following are examples of algebraic polynomials.

${{(a+b)}^{3}}, {{a}^{2}}+{{b}^{2}}+2ab, {{x}^{2}}+{{y}^{2}}+2\sqrt{3}$


What are the Types of Algebraic Polynomials in One Variable?

By the degree of the polynomial, the following are common types of polynomials in one variable.


1. Linear Polynomial

This polynomial is of degree $1$. The standard form of the linear polynomial is ${{a}_{0}}x+{{a}_{1}}\text{;}$

$\text{where }{{a}_{n}}\text{ is a real number;} $

$ n=0,1\text{ ;} $

$ {{a}_{0}}\ne 0$


2. Quadratic Polynomial

This polynomial is of degree $2$. The standard form of the quadratic polynomial is

$ {{a}_{0}}{{x}^{2}}+{{a}_{1}}x+{{a}_{2}}\text{; }$

$ \text{where }{{a}_{n}}\text{ is a real number;}$

$n=0,1,2\text{ ;}$

${{a}_{0}}\ne 0$


3. Cubic Polynomial

This polynomial is of degree $3$. The standard form of the cubic polynomial is ${{a}_{0}}{{x}^{3}}+{{a}_{1}}{{x}^{2}}+{{a}_{2}}x+{{a}_{3}};$

$\text{where }{{a}_{n}}\text{ is a real number;}$

$ n=0,1,2,3\text{ ;}$

$ {{a}_{0}}\ne 0 $


Chart of Useful Algebraic Identities

The following table shows some of the important algebraic identities.


Algebraic Identities

Description

Formula

Whole square of the sum of any two variables

${{(x+y)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$

Whole square of the difference between any two variables.

${{(x-y)}^{2}}={{x}^{2}}+{{y}^{2}}-2xy$

Difference between any two squares

${{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right)$

Whole square of the sum of any three variables

${{(x+y+c)}^{2}}={{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2xy+2yz+2zx$

Whole cube of the sum of any two variables

${{(x+y)}^{3}}={{x}^{3}}+{{y}^{3}}+3x{{y}^{2}}+3{{x}^{2}}y$

Whole cube of the difference between any two variables.

${{(x-y)}^{3}}={{x}^{3}}-3{{x}^{2}}y+3x{{y}^{2}}-{{y}^{3}}$

Sum of any two zubes

${{x}^{3}}+{{y}^{3}}=\left( x+y \right)\left( {{x}^{2}}-xy+{{y}^{2}} \right)$

Difference between any two cubes

${{x}^{3}}-{{y}^{3}}=\left( x-y \right)\left( {{x}^{2}}+xy+{{y}^{2}} \right)$

Whole cube of the sum of any three variables

${{(x+y+z)}^{3}}={{x}^{3}}+{{y}^{3}}+{{z}^{3}}+3\left( x+y \right)\left( y+z \right)\left( z+x \right)$


Interesting Facts

  • The mathematical identity ${{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1$ is not an “algebraic identity” but a “trigonometric identity”.

  • The algebraic identities may be conditional also. For example, ${{a}^{3}}+{{b}^{3}}+{{c}^{3}}=3abc,\quad$ when $ a+b+c=0$

  • The mathematical expression $\sqrt{3}+2$ is not an “algebraic expression” but a numeric expression.


Solved Examples

1. Find the value of $\left( {x{\rm{ }} + {\rm{ }}1} \right)\left( {x{\rm{ }} + {\rm{ }}1} \right)$ using standard algebraic identities.


Ans. Note that $\left( {x{\rm{ }} + {\rm{ }}1} \right)\left( {x{\rm{ }} + {\rm{ }}1} \right)$ is equivalent to ${\left( {x{\rm{ }} + {\rm{ }}1} \right)^2}$.


Thus, by using the Identity applicable for the whole square of the sum of any two variables ${{(x+y)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$ and substituting here $y=1$, we get the following.


${{\left( x+1 \right)}^{2}}\ ={{x}^{2}}\ +2\times x\times 1+{{1}^{2\;}}={{x}^{2\;}}+2x+1$

2. Factorize $\left( {{x^4}\;-{\rm{ }}1} \right)$ using standard algebraic identities.


Ans. Split $\left( {{x^4}\;-{\rm{ }}1} \right)$using the Identity for the difference between any two squares

Replace here x by and put y = 1.

$\left( {{x^4}\;-{\rm{ }}1} \right){\rm{ }} = {\rm{ }}\left( {{{\left( {{x^2}} \right)}^2}-{\rm{ }}{1^2}} \right){\rm{ }} = {\rm{ }}\left( {{x^{2\;}} + {\rm{ }}1} \right)\left( {{x^{2\;}}-{\rm{ }}1} \right)$

The factor $(x^2 -1)$ is further factorized using the same Identity where y = 1, and we obtain the following

$\left( {{x^4}\;-{\rm{ }}1} \right){\rm{ }} = {\rm{ }}\left( {{x^{2\;}} + {\rm{ }}1} \right)\left( {{{\left( x \right)}^{2\;}}-{{\left( 1 \right)}^2}} \right){\rm{ }} = {\rm{ }}\left( {{x^{2\;}} + {\rm{ }}1} \right)\left( {x{\rm{ }} + {\rm{ }}1} \right)\left( {x{\rm{ }}-{\rm{ }}1} \right)$

3. Expand ${\left( {3x{\rm{ }}-{\rm{ }}4y} \right)^{3\;}}$using standard algebraic identities.


Ans: We can expand ${\left( {3x{\rm{ }}-{\rm{ }}4y} \right)^{3\;}}$using the identity applicable to the whole cube of the difference of any two variables ${{(x-y)}^{3}}={{x}^{3}}-3{{x}^{2}}y+3x{{y}^{2}}-{{y}^{3}}$

Replacing x and y by $3x$ and $4y$, we get the following.


${{\left( 3x-4y \right)}^{3}}\ ={{\left( 3x \right)}^{3}}-3{{\left( 3x \right)}^{2}}\left( 4y \right)+3\left( 3x \right){{\left( 4y \right)}^{2}}-{{\left( 4y \right)}^{3}}=27{{x}^{3}}\ -108{{x}^{2}}y+144x{{y}^{2}}\ -64{{y}^{3}}$


4. Factorize ${\rm{ }}({x^3}\; + {\rm{ }}8{y^{3\;}} + {\rm{ }}27{z^3}\;-{\rm{ }}18xyz)$using standard algebraic identities.


Ans. The expression ${\rm{ }}({x^3}\; + {\rm{ }}8{y^{3\;}} + {\rm{ }}27{z^3}\;-{\rm{ }}18xyz)$ can be factorized using the identity \[\left( {{x}^{3}}\ +{{y}^{3\;}}+{{z}^{3}}\ -3xyz \right)=\left( x+y+z \right)\left( {{x}^{2\;}}+{{y}^{2}}\ +{{z}^{2}}\ -xy-yz-zx \right)\]


Therefore, we get the following

$\left( {{x^3}\; + {\rm{ }}8{y^{3\;}} + {\rm{ }}27{z^3}\;-{\rm{ }}18xyz} \right){\rm{ }} = {\rm{ }}{\left( x \right)^3}\; + {\rm{ }}{\left( {2y} \right)^{3\;}} + {\rm{ }}{\left( {3z} \right)^3}\;-{\rm{ }}3\left( x \right)\left( {2y} \right)\left( {3z} \right) = {\rm{ }}\left( {x{\rm{ }} + {\rm{ }}2y{\rm{ }} + {\rm{ }}3z} \right)\left( {{x^{2\;}} + {\rm{ }}4{y^2}\; + {\rm{ }}9{z^2}\;-{\rm{ }}2xy{\rm{ }}-{\rm{ }}6yz{\rm{ }}-{\rm{ }}3zx} \right)$


Summary

  • The standard algebraic format encompasses the algebraic expressions, the algebraic identities, and the algebraic equations.

  • The algebraic expressions, the algebraic identities, and the algebraic equations can be of various types varying from monomials to polynomials.

  • The algebraic expressions, the algebraic identities, and the algebraic equations can be linear, quadratic, cubic, or of any higher degree.

  • The algebraic identities find usages as algebraic formulae in various mathematical problems.


Practice Problems

1. Find the value of ${{\left( a+2 \right)}^{3}}$.


2. Find the value of ${{\left( a+2+3x \right)}^{2}}$.


3. Find the value of $729{{x}^{3}}-216{{y}^{3}}$

FAQs on Standard Algebraic Formats

1. What is the basic difference between an algebraic expression and an algebraic equation?

An algebraic expression involves variables, arithmetic operators, and constants. On the other hand, an algebraic equation is what you get on writing two algebraic expressions using the "equal to" sign or relating an algebraic expression to any constant.

2. What is the basic difference between an algebraic identity and an algebraic equation?

An algebraic identity is true for all the values of its variables.

An algebraic equation, if solvable, has a finite number of solutions.

3. What is the associative law of algebraic operations?

The Associative law of algebraic operations applies to addition, subtraction, division, or multiplication, as shown below.


$x+\left( y+z \right)\text{ }=\text{ }\left( x+y \right)+z\text{ }$ 

$ x\times \left( y\times z \right)\text{ }=\text{ }\left( x\times y \right)\times z$.