Introduction to Algebra: Definition, Properties, Examples - Embibe
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  • Written By Madhurima Das
  • Last Modified 24-06-2022
  • Written By Madhurima Das
  • Last Modified 24-06-2022

Introduction to Algebra: Definition, Properties, Examples

We come across many situations where the value of a constant keeps changing over time. So, we use a variable to store the continuous value when it changes. Algebra is one of the most important branches of mathematics where we introduce the concept of the variables. We use algebra in higher mathematics also. We perform arithmetic operations using letters in algebra.

We can apply algebra in our real-life situations using algebraic expressions and equations. In this article, let’s learn about the basic things about algebra and its applications.

What is Algebra?

Algebra is generalized arithmetic in which we represent the numbers by letters, known as literals numbers or simply literals. These letters do not have any fixed values and are called variables. In our real-life scenario, we see some values that keep on varying. But there is a constant need to represent these fluctuating values.

Learn All Concepts on Algebraic Expressions

Here in algebra, we depict these values with letters such as \(a,b,c,x,y,z,p,\) or \(q,\) and these letters are called variables. Further, these letters undergo various arithmetic operations of addition, subtraction, multiplication, and division to find the values. Using these arithmetic operations and the letters, we can form algebraic expressions.

Branches of Algebra

Algebra can be categorized into different branches that are showing below:

  1. Pre-algebra 
  2. Elementary Algebra
  3. Abstract Algebra
  4. Universal Algebra

Pre-algebra 

The fundamental ways of expressing the unknown values as variables make it easier to create mathematical expressions. It helps in converting real-life problems into an algebraic expression in mathematics. Developing a mathematical expression for any problem statement is a component of pre-algebra.

Elementary Algebra

In elementary algebra, variables like \(x,y,\) are presented in the form of an equation. Based on the highest degree of the variable, the equations are called linear equations, quadratic equations, cubic equations, etc.

We can write the linear equations in the form of

\(ax + b = c\)
\(ax + by + c = 0\)
\(ax + by + cz + d = 0\)
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Based on the degree of the variables, they branch out into quadratic equations and polynomials.

A standard form of representation of a quadratic equation is \(a{x^2} + bx + c = 0.\)
The standard for of representation of a cubic equation is \(a{x^3} + b{x^2} + cx + d = 0.\)

Abstract Algebra

Abstract algebra operates using abstract concepts like rings, groups, vectors rather than simple mathematical number systems. Ring theory and group theory are two important areas of abstract algebra of higher mathematics. Abstract algebra finds various applications in physics, astronomy, computer science, etc.

Universal Algebra

All the other mathematical forms involving calculus, trigonometry, coordinate geometry involving algebraic expressions can be described as universal algebra. Other branches of algebra can be counted as the subset of universal algebra. Any real-life problem can be classified into one of the branches of mathematics and can be solved using abstract algebra.

Algebraic Expression

The combination of the constants and the variables connected by some or all of the four fundamental operations addition \(\left( + \right),\) subtraction \(\left( – \right),\) multiplication \(\left( \times \right)\) and division \(\left( \div \right),\) is known as an algebraic expression.
Examples: \(2x + 5,y – 5,4x + 5,3y – 5,\) etc.

Operation on Literals and Constants

Addition

1. The sum of \(x\) and \(6\) can be written as \(x + 6.\)
2. \(a\) more than \(b\) can be written as \(a + b.\)

Subtraction

1. \(4\) less than \(x\) can be written as \(x – 4.\)
2. \(a\) less than \(b\) can be written as \(b – a.\)

Multiplication

1. \(6\) times \(x\) can be written as \(6x.\)
2. The product of \(a\) and \(b\) can be written as \(ab\) or \(ba.\)

Division

1. \(x\) divided by \(y\) can be written as \(\frac{x}{y}.\)
2. \(a\) divided by \(10\) can be written as \(\frac{a}{{10}}.\)

Terms

In an algebraic expression, a term may consist of only constant, only one variable, the product of two or more variables, a product of both the variable and the constant part. The terms may be positive or negative. Example: \(5, – 11,x,y,xy,yz,\) are the terms.
1. Here, \(5,\) is a positive constant term.
2. \(-11\) is a negative constant term.
3. \(x,y\) are positive terms with one variable only.
4. \( – x, – y\) are negative terms with one variable only.
5. \(xy,yz\) are positive terms with the product of two variables.

Coefficients

In a product of numbers and literals, any of the factors is called the coefficient of the product of the factors.
Example: In the term \(2xyz,\) the numeral coefficient is \(2,\) coefficient of \(x\) is \(2yz,\) coefficient of \(y\) is \(2xz,\) etc.
The coefficients can be of two types. One is the numeral coefficient, and the other one is the literal coefficient.
Let us take an example and know about this.
In the algebraic term \(5xy,5\) is the numeral coefficient of \(xy,\) and \(xy\) is the literal coefficient of \(5.\)

Algebraic Expressions Based on the Number of the Terms

1. Monomial: It is an algebraic expression that contains only one term. Example: \(5x,2xy, – 3{a^2}b, – 7\) etc., are monomials.
2. Binomials: An algebraic expression that contains two terms is known as binomial. Example: \(\left({2a + 3b} \right),\left({8 – 3x} \right),\left({{x^2} – 4x{y^2}} \right)\) etc., are binomials.
3. Trinomials: An algebraic expression that has three terms is known as trinomial. Example: \(\left({a + 2b + 5c} \right),\left({x + 2y – 3z} \right),\left({{x^3} – {y^3} – {z^3}} \right)\) etc., are trinomials.
4. Quadrinomials: An algebraic expression including four terms is known as quadrinomial. Example:\(\left({x + y + z – 5} \right),\left({{x^3} + {y^3} + {z^3} + 3xyz} \right),\) etc., are quadrinomials.

Polynomials

An algebraic expression including two or more terms with the exponents of the variables as whole numbers is known as a polynomial. It includes all the Algebraic expressions with one or more terms and binomial, trinomial, quadrinomials, etc.

Algebraic Identities

If an equation is valid for all values of the variables in it, it is called an identity. Algebraic identities are the equation in which the value of the left-hand side of an equation identically equals the value of the right-hand side of the equation.
Example: Consider the linear equation \(ax + b = 0.\)
Here, the left-hand side and right-hand sides of the above equations are the same when \(x = \, – \frac{b}{a}.\)

Standard Algebraic Identities

We have some standard identities to use in the various branches of mathematics. All the standard identities are derived by using the Binomial theorem.

Four standard algebraic identities are listed below:

Identity 1: Algebraic identity of the square of the sum of two terms

\({\left({a + b} \right)^2} = {a^2} + 2ab + {b^2}\)

Identity 2: Algebraic identity of the square of the difference of two terms

\({\left({a – b} \right)^2} = {a^2} – 2ab + {b^2}\)

Identity 3: Algebraic identity of the difference of two squares

\(\left({a + b} \right)\left({a – b} \right) ={a^2} – {b^2}\)

Identity-4: Algebraic Identity \(\left({x + a}\right)\left({x + b} \right)\)

\(\left({x + a} \right)\left({x + b} \right) = {x^2} + \left({a + b} \right)x + ab\)

Algebraic Equation

A condition on a variable is called an equation. The conditions are

  1. two expressions should have an equal value
  2. out of the two expressions, at least one must contain the variable

An equation remains the same when the expressions on the left and the right are interchanged.

For example, \(x + 2y = 0,a + 3b + c = 0,2{x^2} + 5 = 0\) are the equations.

Solved Examples – Introduction to Algebra

Q.1. Add \(5{x^2} – 7x + 3, – 8{x^2} + 2x – 5\) and \(7{x^2} – x – 2.\)
Ans:
The solution is given below
\( = \left({5{x^2} – 7x + 3} \right) + \left({ – 8{x^2} + 2x – 5}\right) + \left({7{x^2} – x – 2} \right)\)
\( = 5{x^2} – 8{x^2} + 7{x^2} – 7x + 2x – x + 3 – 5 – 2\) (collecting like terms)
\( = \left({5 – 8 + 7} \right){x^2} + \left({ – 7 + 2 – 1} \right)x + \left({3 – 5 – 2} \right)\) (adding like terms)
\( = 4{x^2} – 6x – 4\)

Q.2. Subtract \(\left({2{x^2} – 5x + 7} \right)\) from \(\left({3{x^2} + 4x – 6} \right).\)
Ans: \(\left({3{x^2} + 4x – 6} \right) – \left({2{x^2} – 5x + 7} \right)\)
\( = 3{x^2} + 4x – 6 – 2{x^2} + 5x – 7\)
\( = \left({3 – 2} \right){x^2} + \left({4 + 5} \right)x + \left({ – 6 – 7} \right)\)
\( = {x^2} + 9x – 13\)

Q.3. Multiply: \( – 8a{b^2}c,3{a^2}b\) and \( – \frac{1}{6}.\)
Ans: \(\left({ – 8a{b^2}c}\right) \times \left({3{a^2}b}\right) \times \left({ – \frac{1}{6}}\right)\)
\( = \left({ – 8 \times 3 \times \frac{{ – 1}}{6}} \right) \times \left({a \times {a^2} \times {b^2} \times b \times c} \right)\)
\( = 4{a^{\left({1 + 2}\right)}} \times {b^{\left({2 + 1} \right)}} \times c = 4{a^3}{b^3}c\)

Q.4. Find the product of \(\left({x + 2} \right)\left({x + 2} \right)\) using standard algebraic identities.
Ans: We can write \(\left({x + 2} \right)\left({x + 2} \right)\) as \({(x + 2)^2}.\)
By using the identity: \({\left({a + b} \right)^2} = {a^2} + 2ab + {b^2}\)
By putting the value of \(x = a\) and \(y = 2,\)We get
\( \Rightarrow {\left({x + 2} \right)^2} = {x^2} + {2^2} + 2 \times x \times 2\)
\( \Rightarrow {\left({x + 2} \right)^2} = {x^2} + 4 + 4x\)
Hence, the value of \(\left({x + 2} \right)\left({x + 2}\right)\) is \({x^2} + 4x + 4.\)

Q.5. Solve the equation \(2x + 9 = 19.\)
Ans: The given equation is \(2x + 9 = 19.\)
Now, transposing \(9\) to RHS we have,
\( \Rightarrow 2x = 19 – 9\)
\( \Rightarrow 2x = 10\)
Now, divide both sides by \(2.\)
\( \Rightarrow \frac{{2x}}{2} = \frac{{10}}{2}\)
\( \Rightarrow x = 5\)
Hence, the solution of equation \(2x + 9 = 19\) is \(x = 5\)

Summary

In this article, we have learned about basic algebra. We have discussed the branches of algebra in brief, algebraic expressions, algebraic identities, and algebraic equations. We solved some problems related to algebra.

List of Important Algebra Formulas

Frequently Asked Questions (FAQs) – Introduction to Algebra

Q.1. What are the basics of algebra?
Ans: The basics of algebra consist of numbers, variables, constants, expressions, equations, quadratic equations, linear equations; further, it involves the basic arithmetic operations of addition, subtraction, multiplication, and division within the algebraic expressions.

Q.2. What are the four rules of algebra?
Ans:
There are four basic rules or properties of algebra viz. commutative, associative, distributive, and identity. 

Q.3. What is the golden rule for solving algebraic equations?
Ans: In an equation, an equality sign is always there. The equality sign shows that the expression’s value in the left of the equal sign is equal to the value of the expression to the right of the equal sign. So we can have an equation using variables, algebraic expressions with an equality sign.
We will keep the variable terms on the LHS of the equal sign and the constant terms on the RHS to find the unknown variable.

Q.4. What is algebra formula?
Ans: The different formulae used in solving problems related to algebraic expressions is called algebra formula. Some of them are:
1. \({a^2} – {b^2} = \left({a – b}\right)\left({a + b} \right)\)
2. \({\left({a + b} \right)^2} = {a^2} + 2ab + {b^2}\)
3. \({\left({a – b} \right)^2} = {a^2} – 2ab + {b^2}\)
4. \({\left({a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca\)
5. \({\left({a + b} \right)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} = {a^3} + {b^3} + 3ab\left({a + b} \right)\)
6. \({\left({a – b} \right)^3} = {a^3} – 3{a^2}b + 3a{b^2} -{b^3} = {a^3} – {b^3} – 3ab\left({a – b} \right)\)
7. \({a^3} – {b^3} = \left({a – b} \right)\left({{a^2} + ab + {b^2}} \right)\)
8. \({a^3} + {b^3} = \left({a + b} \right)\left({{a^2} – ab + {b^2}} \right)\)
9. \({a^3} + {b^3} + {c^3} – 3abc = \left({a + b + c} \right)\left({{a^2} + {b^2} + {c^2} – ab – bc – ca} \right)\)

Q.5. How to simplify an algebraic expression?
Ans: The combination of the constants and the variables that are linked by the signs of fundamental operations of addition, subtraction, multiplication, and division is called an algebraic expression. To simplify an algebraic expression, we need to regroup the like terms and perform the fundamental operations.

We hope this detailed article on introduction to algebra helped you in your studies. If you have doubts or queries regarding this topic, feel free to ask us in the comment section. Happy learning!

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