Algebraic Expressions Worksheets

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# Algebraic Expressions Worksheets

Edited By Team Careers360 | Updated on Jan 29, 2024 05:26 PM IST

Two or more algebraic expressions use the comparator signs among themselves to become an algebraic equation or an algebraic identity. There are various algebraic identities that are helpful in solving problems on simplifications and expansions. All these algebraic standard formats display the commonly used mathematical operators. They represent the mathematical statements in terms of the common operations of division, multiplication, addition, subtraction, and exponentiation. For example, 5{{x}^{2}}-2xy+c , is an algebraic expression.

Let us explore this article to realize the concept of “algebraic expressions” clearly.

## What Are Mathematical Expressions?

The term “mathematical expression” implies a symbolic presentation of any valid mathematical statement, which may include any combination of mathematical operations like division, multiplication, addition, subtraction, and exponentiation as indicated by the corresponding mathematical operators.

The following are examples of mathematical expressions.

\begin{aligned}

& 1) \sqrt{3}+2 \\

& 2) a x^2+b \\

& 3) y^3+3

\end{aligned}

## What Are Algebraic Expressions?

The mathematical expression that contains “algebraic Variables”, “coefficients”, and “constants” in any combination is called an “algebraic expression.

Get a clear idea with the following examples.

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

• The mathematical expressions $a{{x}^{2}}+b,\ \ {{y}^{3}}+3$ are all “algebraic expressions”.

## What Are Equations?

Suppose, you use the “equal to symbol” represented as “=” to unite any two expressions. The resulting mathematical entity is what you call an “equation”. An equation, basically, tells that two expressions on either side of the “=” are equivalent.

The following are examples of equations.

\begin{aligned}

&1) \sqrt{3} \tan \theta+2=\cos \alpha \\

& 2) a x^2+b=c \\

& 3) y^3=3 \\

& 4) \cos ^3 \theta+3=\sin ^2 \theta

\end{aligned}

## What Are Algebraic Equations?

Suppose, you use the “equal to sign” symbolized as “=” to unite any two algebraic expressions or an algebraic expression to any constant. The subsequent mathematical entity is what you declare an “algebraic equation”. An algebraic equation, actually suggests that the terms on either side of the “=” are equivalent.

Get a clear idea with the next examples.

• The mathematical equations $\sqrt{3}\tan \theta +2=\cos \alpha ,\ \ {{\cos }^{3}}\theta +3={{\sin }^{2}}\theta$ are not “algebraic equations” but are “trigonometric equations”.

• The mathematical equations $a{{x}^{2}}+b=c,\ \ {{y}^{3}}=3$ are all “algebraic equations”.

## What Are Identities?

Identities are those mathematical entities which are true for any value of the variables involved in one or both the expressions on either side of the “=”.

Following are examples of identities.

\begin{aligned}

& (a+b)^2=a^2+b^2+2 a b \\

& \cos ^2 \theta+\sin ^2 \theta=1

\end{aligned}

## What Are Algebraic Identities?

The identities which involve only the algebraic expressions and may contain coefficients of variables and constants are called “algebraic identities”.

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

Get a perfect idea with the following examples.

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

• The mathematical identity ${{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ is an “algebraic identity”.

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## What Are The Properties of Algebraic Identities?

The following are some of the properties of algebraic identities.

• An algebraic identity does not have a finite number of solutions, as it holds good for an infinite number of values that you put in place of its variables.

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

• The algebraic identities may be used in the factorization of algebraic expression

• The algebraic identities help in solving any algebraic equation.

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

## What Are Algebraic Polynomials?

Algebraic polynomials are those expressions or equations which involve 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.

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

## Chart Of Identities Of Factorization Of Polynomials

As you can exploit the algebraic identities both to simplify and factorize the algebraic expressions, here is a quick reference to some of the algebraic identities that you can use for factorization of polynomials.

 Algebraic Identities Description Formulae Whole square of the sum of any two variables ${{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ Whole square of the difference between any two variables. ${{(a-b)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$ Difference between any two squares ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ Whole square of the sum of any three variables ${{(a+b+c)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ca$ Whole cube of the sum of any two variables ${{(a+b)}^{3}}={{a}^{3}}+{{b}^{3}}+3a{{b}^{2}}+3{{a}^{2}}b$ Whole cube of the difference between any two variables. ${{(a-b)}^{3}}={{a}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}-{{b}^{3}}$ Sum of any two cubes ${{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}-ab+{{b}^{2}} \right)$ Difference between any two cubes ${{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)$ Whole cube of the sum of any three variables ${{(a+b+c)}^{3}}={{a}^{3}}+{{b}^{3}}+{{c}^{3}}+3\left( a+b \right)\left( b+c \right)\left( c+a \right)$

## Chart Of Algebraic Identities

The following are some of the algebraic identities that you will find useful for simplification of algebraic terms or factorization of an algebraic expression or solving any algebraic equation.

 Algebraic Identities $(x+a)(x+b)={{x}^{2}}+\left( a+b \right)x+ab$ ${{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca=\frac{1}{2}\left[ {{\left( a-b \right)}^{2}}+{{\left( b-c \right)}^{2}}+{{\left( c-a \right)}^{2}} \right]$ ${{a}^{4}}-{{b}^{4}}=\left( {{a}^{2}}+{{b}^{2}} \right)\left( a+b \right)\left( a-b \right)$ ${{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=\frac{1}{2}(a+b+c)\left[ {{\left( a-b \right)}^{2}}+{{\left( b-c \right)}^{2}}+{{\left( c-a \right)}^{2}} \right]$ ${{a}^{4}}+{{a}^{2}}+1=\left( {{a}^{2}}+a+1 \right)\left( {{a}^{2}}-a+1 \right)$ \\ a^3+b^3+c^3=3 a b c, when \ \ a+b+c=0 \\ a^3+b^3+c^3 \neq 3 a b c, when \ \ a+b+c \neq 0 $a-b=\left( \sqrt{a}+\sqrt{b} \right)\left( \sqrt{a}-\sqrt{b} \right),\ \ for\ a,b>0$

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 are the laws of algebraic operations?

The basic rules of algebraic operations, which include addition, subtraction, division multiplication are as follows:

• Associative law

\begin{aligned}

& x+(y+z)=(x+y)+z \\

& x \times(y \times z)=(x \times y) \times z

\end{aligned}

• Commutativity law

\begin{aligned}

& x+y=y+x \\

& y \times z=z \times y

\end{aligned}

3. How an algebraic identity can be solved?

As such an algebraic identity cannot be solved. An algebraic identity can have an infinite number of values that you put in place of its variables. This is the essence of an algebraic identity that it is always true for any value of the variable.

4. What are the different types of identities?

Some of the different types of identities are

• Algebraic Identities

• Trigonometric Identities

• Mensuration Identities

• Geometric Identities

5. 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.

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##### Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available
##### Automation Test Engineer

An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.

2 Jobs Available