Algebraic Expressions Questions

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# Algebraic Expressions Questions

Edited By Team Careers360 | Updated on Jan 29, 2024 05:47 PM IST

The concept of “algebraic identities” will be clear to you, once you have an idea about the defining characteristics of “expressions”, “identities”, etc. The terms “algebraic expressions”, “algebraic equations”, and “algebraic identities” have the common word “algebraic”.

The fact is that all “mathematical expressions” are not “algebraic expressions”. Likewise, “identities” encompass the “algebraic identities” along with the other types of identities. Thus it is important to identify when an “expression” becomes an “algebraic expression”, and when an “equation” becomes an “algebraic equation”. You will here, explore the difference between an “identity” and an “algebraic identity”. Also, you will get an idea about the functionalities of “algebraic identities”

It is presumed that the terms “constants”, “Variables”, “and coefficients”, and mathematical operations like multiplication, division, subtraction, addition, and exponentiation are known to you.

## what Are The Types Of Mathematical Expressions?

As such the symbolic arrangement of any combination of mathematical operations like addition, division, multiplication, subtraction, and exponentiation as specified by the corresponding mathematical operators, as per any valid mathematical statement, is termed the “mathematical expression”.

Following are some of the different types of mathematical expressions.

• Numeric Expressions: The mathematical expression that consists of no “Variables” and hence no “coefficients”, but only numerical values or “constants” is called a “numeric expression. The mathematical expression $\frac{\sqrt{3}}{4}+2$ is a numeric expression.

• Algebraic Expressions: The mathematical expression that contains “algebraic Variables”, “coefficients”, “constants” in any combination is called an “algebraic expression.

Note that the algebraic expressions”. $2{{x}^{3}}+a,\ \ {{y}^{2}}+3$

• Trigonometric Expressions: The mathematical expression that has “trigonometric terms” in “Variables”, “constants” in any combination is called a “trigonometric expression.

See that the mathematical expressions $2{{\cos }^{3}}\theta +a,\ \ {{\cos }^{2}}\frac{\pi }{2}+3$ are “trigonometric expressions”.

## What Are The Types Of Equations?

An “equation” has the “equal to symbol” represented as “=” to join any two expressions or an algebraic expression to any constant. An algebraic equation, essentially, infers that the mathematical terms on either side of the “=” symbol are equivalent.

Following are some of the different types of equations.

• Algebraic Equations: The mathematical equation that contains “algebraic Variables” with or without “coefficients”, equal to other algebraic variables or “constants” in any combination is called an “algebraic Equation. Note that the mathematical equations $\frac{2}{5}{{x}^{3}}+a{{y}^{2}}=b,\ \ 5{{y}^{2}}+3=0$ are “algebraic Equations”.

• Trigonometric Equations: The mathematical equation that has “trigonometric terms” in “Variables”, “constants” in any combination is called a “trigonometric Equation.

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See that the mathematical Equations $2{{\cos }^{3}}\theta +a=b,\ \ 4{{\cos }^{2}}\frac{\pi }{2}-1=\sin \alpha$ are “trigonometric Equations”.

## What Are The Types Of Identities?

You name those mathematical entities as “Identities” which are correct for any value of the variables present in one or both the expressions on either side of the “=”.

Following are some of the different types of Identities.

• Algebraic Identities: The identity involving only the algebraic expressions or an algebraic expression and any constant, such that on putting any value in the variable, the equality holds good, is called an algebraic identity.

Note that these “algebraic Identities”. ${{(x+y)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy,\ \ {{(x-y)}^{2}}={{x}^{2}}+{{y}^{2}}-2xy$

• Trigonometric Identities: The identity that has “trigonometric terms” in “Variables”, “constants” in any combination is called a “trigonometric Equation.

See that these trigonometric Identities.${{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1,\ \ 1+{{\tan }^{2}}\theta ={{\sec }^{2}}\theta$.

• Mensuration Identities: The identity that has “mensuration terms” in “Variables”, and “constants” in any combination is called a “mensuration Equation.

Note the following “mensuration Identities”.\begin{align} {{a}^{2}}+{{b}^{2}}={{c}^{2}},\ \ \text{for any right angled triangle} \\ area=sid{{e}^{2}},\ \ \text{for any square} \\ \end{align}

### What Are The Algebraic Identities Important?

The algebraic identities are important from the mathematical point of view and the following are some of the reasons.

• An algebraic identity cannot have a finite number of solutions, as it is true for an infinite number of values that you put in place of its variables.

• For the simplification of algebraic polynomials, they are useful.

• They are beneficial in the factorization of algebraic expression

• They find applications in solving any algebraic equation.

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

## What Are Algebraic Polynomials?

You may come across expressions or equations which involve only the non-negative integral power of variables but may have any integral coefficients and constants. These algebraic entities are called algebraic polynomials

The examples of algebraic polynomials are the following.

${{(a+b)}^{4}},\ \ {{p}^{2}}+{{q}^{2}}+2ab,\ \ {{a}^{2}}+{{b}^{2}}+2\sqrt{3}$

Chart Of Identities Of Factorization Of Polynomials

As you find the algebraic identities useful to factorize, and simplify the algebraic expressions, here is your quick reference chart for some of the algebraic identities that you will use for factorization of the polynomials, whenever necessary.

 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

For simplification of algebraic terms, solving any algebraic equation or factorization of an algebraic expression, the following are some of the useful algebraic identities.

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

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