Order of Operations on Algebraic Expressions: Definition, BODMAS

Order of Operations on Algebraic Expressions: Uptill now, we were studying arithmetic in which you were mostly using arithmetic symbols \(0,\,1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9\). These arithmetic symbols are also known as numerals. We also used signs \( + \) for addition, \( – \) for subtraction, \( \times \) for multiplication and \( \div \) for the division to do arithmetical calculations.

Now, we are going to learn algebra. In this article, we will first study basic operations of addition, subtraction, multiplication and division on letter symbols. Then we shall also discuss the idea of an algebraic expression, its terms, evaluation, etc. The addition and subtraction of algebraic expressions will also be discussed.

Algebra

Besides the arithmetic symbols (or numerals) and signs for the operations like addition, subtraction, multiplication and division, algebra also uses letters from various alphabets like English, Greek etc. These letters are used to generalise results or represent unknown quantities known as letter symbols, literals, or variables.

Learn the Concepts of Algebraic Expressions

Algebraic Expressions

A combination of constants and variables connected by the signs of fundamental operations of addition, subtraction, multiplication and division is called an algebraic expression.

Various parts of an algebraic expression separated by the signs \( + \) or \( – \) are called the terms of the expression.

Example: \(5x + 4y – z\) is an algebraic expression having \(5x,\,4y\) and \( – z\) as its terms.

Types of Algebraic Expressions

Monomial: A monomial is an algebraic expression containing only one term.

Example: \(2,\,3x,\,4y,\,6{x^2}y\) etc.

Binomial: A binomial is an algebraic expression containing only two terms.

Example: \(4 + x,\,5 – y,\,{b^2} – 6,\,{x^3} + 8\) etc.

Trinomial: A trinomial is an algebraic expression containing only three terms.

Example: \(a + b + c,\,3x + 4y – z,\,{x^2} + 2x + y\) etc

Quadrinomial: A quadrinomial is an algebraic expression containing only four terms.

Example: \({a^3} + {b^3} + {c^3} + 3abc,\,{x^2} + {y^2} + {z^2} + 5\)

Polynomial: A polynomial is an algebraic expression containing one or more terms with powers of variables as only whole numbers.

Example: \(3x + 6y + 4,\,{a^2} – 6\) etc.

Factors and Coefficients

Factors are numbers and literals multiplied together to get the given expression. For example, \(14xyz\) is the product of \(14,\,x,\,y\) and \(z\).

Any factor or group of factors of a term is called the coefficient of the remaining part of the term. For example, in the term \(9xy\), the numerical coefficient is \(x\) while the literal coefficient of the term is \(xy\). Also, the coefficient of \(x\) is \(9y\) and the coefficient of \(y\) is \(9x\).

Like and Unlike Terms

Terms of an expression with the same literal coefficient are called like terms. Else they are called, unlike terms. For example, \(8xy{z^2},\,2xy{z^2},\,3xy{z^2}\) are like terms and \(5{a^2},\,3{b^2},\,6{a^2}{b^2}\) are all unlike terms.

Operations on Algebraic Expressions

To add or subtract two polynomials, arrange the like terms together and simplify them according to their signs. The sum of the like terms is another like term whose coefficient is the sum of the coefficients of the terms added. Note that unlike terms cannot be simplified further and have to be written as they are.

Addition of positive like terms:

To add several positive like terms, we proceed as follows:

1. Obtain all like terms.
2. Find the sum of the numerical coefficients of all terms.
3. Write the required sum as a like term whose numerical coefficient is the numerical obtained in step \(2\) and literal factor is the same as the literal factors of the given like terms. 

Addition of negative like terms:

To add negative like terms, we proceed as follows:

1. Obtain all like terms.
2. Obtain the sum of the numerical coefficients (without negative sign) of all like terms.
3. Write an expression as a product of the number obtained in step \(2\), with all the literal coefficients preceded by a minus sign.
4. The expression obtained in step \(3\) is the required result.

Addition of positive and negative like terms:

To add positive and negative like terms, we proceed as follows:

1. Collect all positive like terms and find their sum.
2. Collect all negative like terms and find their sum.
3. Obtain the numerical coefficients (without negative signs) of like terms obtained in steps \(1\) and \(2\).
4. Subtract the numerical coefficient in step \(2\) from the numerical coefficient in step \(1\). Write the answer as a product of this number and all the literal coefficients.

Subtraction of algebraic expression: To subtract an algebraic expression from another, we should change the signs (from \( + \) to \( – \) or from \( – \)  to \( + \)) of all the terms of the expression, which is to be subtracted and then the two expressions are added.

Example 1: Add \(5{x^3} + 4{y^2}z + 3xy\) and \(3{y^3} + 6{y^2}z + 4xy\)

Solution: \(\left( {5{x^3} + 4{y^2}z + 3xy} \right) + \left( {3{y^3} + 6{y^2}z + 4xy} \right)\)
\( = 5{x^3} + 3{y^3} + 4{y^2}z + 6{y^2}z + 3xy + 4xy\)
\( = 5{x^3} + 3{y^3} + 10{y^2}z + 7xy\)

Thus the required sum is \(5{x^3} + 3{y^3} + 10{y^2}z + 7xy\)

Example 2: Subtract \(12{x^2}yz + 3{x^2} + 7\) from \(9{x^2}yz – 5{x^2} – 11\)

Solution: \(\left( {9{x^2}yz – 5{x^2} – 11} \right) – \left( {12{x^2}yz + 3{x^2} + 7} \right)\)
\( = 9{x^2}yz – 5{x^2} – 11 – 12{x^2}yz – 3{x^2} – 7\)
\( = 9{x^2}yz – 12{x^2}yz – 5{x^2} – 3{x^2} – 11 – 7\)
\( = 3{x^2}yz – 8{x^2} – 18\)

Use of Grouping Symbols (Brackets) in Writing Algebraic Expressions

In dealing with algebraic expressions, sometimes it becomes necessary to consider an expression consisting of two or more terms as a single term. For example, if we say that the sum of \(2x – y + 3\) and \(x + 2y + 1\) are subtracted from the sum of \( +y \) and \(3x – 4y + 5\).

In this case, the sum of \(2x – y + 3\) and \(x + 2y + 1\) is taken as one term, and the sum of \(x + y\) and \(3x – 4y + 5\) is also taken as one term. By using brackets (grouping symbols), the above statement can be written as

\(\left\{ {\left( {2x – y + 3} \right) + \left( {x + 2y + 1} \right)} \right\} – \left\{ {\left( {x + y} \right) + \left( {3x – 4y + 5} \right)} \right\}\)

Thus, we need to insert the brackets (or grouping symbols) to perform algebraic operations.

Removal of Brackets

In the above section, we have seen that when we make operations on two or more algebraic expressions, we use the symbols of groupings, i.e., parentheses, braces and brackets. In simplifying such expressions, we first remove the grouping symbols by using the following rules:  

1. If a \( + \) sign precedes a grouping symbol, the grouping symbol may be removed without any change in the sign of the terms.
2. If a \( – \) sign precedes a symbol of grouping, the grouping symbol may be removed, and the sign of each term is reversed.
3. If more than one grouping symbol is present in an expression, we remove the innermost grouping symbol first and collect and combine like terms, if any. We continue this process outwards until all the grouping symbols have been removed.

Order of Operations on Algebraic Expressions

Before we can evaluate an expression, we need to know how the operations are done.

For all algebraic expressions, the order of evaluation is (BEDMAS ), abbreviated as Brackets, Exponents, Division, Multiplication, Addition and Subtraction.

The order of operations is as follows:

1. Evaluate the brackets, if there are any, and if they required evaluation
2. Evaluate the powers, i.e., exponents
3. Multiply or divide as required.
4. Add or subtract.

Solved Examples – Order of Operations on Algebraic Expressions

Q.1. Simplify \(\left( {{a^2} + {b^2} + 2ab} \right) + \left( {{a^2} + {b^2} – 2ab} \right)\)
Ans: \(\left( {{a^2} + {b^2} + 2ab} \right) + \left( {{a^2} + {b^2} – 2ab} \right)\)
Since \( + \) sign precedes the second parentheses, we remove it as it is
\( \Rightarrow {a^2} + {b^2} + 2ab + {a^2} + {b^2} – 2ab\)
Grouping like terms, we get
\( \Rightarrow {a^2} + {a^2} + {b^2} + {b^2} + 2ab – 2ab\)
\( = 2{a^2} + 2{b^2}\)
\( = 2\left( {{a^2} + {b^2}} \right)\)

Q.2. Simplify \( – 5(a + b) + 2(2a – b) + 4a – 7\)
Ans: \( – 5(a + b) + 2(2a – b) + 4a – 7\)
Removing the first parentheses by multiplying each term inside it with \( -5 \) and the second parentheses by multiplying each term with \(2\) inside it, we get
\( = – 5a – 5b + 4a – 2b + 4a – 7\)
Grouping the like terms, we get
\( = – 5a + 4a + 4a – 5b – 2b – 7\)
\( = 3a – 7b – 7\)
Therefore, \( – 5(a + b) + 2(2a – b) + 4a – 7 = 3a – 7b – 7\)

Q.3. Simplify \(15x – \left[ {8{x^3} + 3{x^2} – \left\{ {8{x^2} – \left( {4 – 2x – {x^3}} \right) – 5{x^3}} \right\} – 2x} \right]\)
Ans: We first simplify the innermost group symbol \({\rm{()}}\), then \(\{ \} \) and then \({\rm{[]}}\). Thus, we have
\(15x – \left[ {8{x^3} + 3{x^2} – \left\{ {8{x^2} – \left( {4 – 2x – {x^3}} \right) – 5{x^3}} \right\} – 2x} \right]\)
\( = 15x – \left[ {8{x^3} + 3{x^2} – \left\{ {8{x^2} – 4 + 2x + {x^3} – 5{x^3}} \right\} – 2x} \right]\)
\( = 15x – \left[ {8{x^3} + 3{x^2} – 8{x^2} + 4 – 2x – {x^3} + 5{x^3} – 2x} \right]\)
\( = 15x – \left[ {12{x^3} – 5{x^2} – 4x + 4} \right]\)
\( = 15x – 12{x^3} + 5{x^2} + 4x – 4\)
\( =  – 12{x^3} + 5{x^2} + 19x – 4\)
Therefore, \(15x – \left[ {8{x^3} + 3{x^2} – \left\{ {8{x^2} – \left( {4 – 2x – {x^3}} \right) – 5{x^3}} \right\} – 2x} \right] =  – 12{x^3} + 5{x^2} + 19x – 4\) 

Q.4. Simplify \(5 + x – \left\{ {2y – \left( {6x + y – 4 + 2{x^2}} \right) – \left( {{x^2} – 2y} \right)} \right\}\)
Ans: We first simplify the innermost group symbol \({\rm{()}}\), and then \(\{ \} \)
\( = 5 + x – \left\{ {2y – 6x – y + 4 – 2{x^2} – {x^2} + 2y} \right\}\)
\( = 5 + x – 2y + 6x + y – 4 + 2{x^2} – {x^2} + 2y\)
Grouping like terms, we get
\( = 2{x^2} – {x^2} + 6x + x – 2y + 2y + 5 – 4\)
\( = {x^2} + 7x + 1\)
Therefore, \(5 + x – \left\{ {2y – \left( {6x + y – 4 + 2{x^2}} \right) – \left( {{x^2} – 2y} \right)} \right\} = {x^2} + 7x + 1\)

Q5. Simplify and find the value of the expression when \(a = 3\) and \(b = 1\)
\(4\left( {{a^2} + {b^2} + 2ab} \right) – \left[ {4\left( {{a^2} + {b^2} – 2ab} \right) – \left\{ { – {b^3} + 4(a – 3)} \right\}} \right]\)
Ans: Proceeding outward from the innermost bracket, we get
\(4\left( {{a^2} + {b^2} + 2ab} \right) – \left[ {4\left( {{a^2} + {b^2} – 2ab} \right) – \left\{ { – {b^3} + 4(a – 3)} \right\}} \right]\)
\( = 4\left( {{a^2} + {b^2} + 2ab} \right) – \left[ {4\left( {{a^2} + {b^2} – 2ab} \right) – \left\{ { – {b^3} + 4a – 12} \right\}} \right]\)
\( = 4{a^2} + 4{b^2} + 8ab – \left[ {4{a^2} + 4{b^2} – 8ab + {b^3} – 4a + 12} \right]\)
\( = 4{a^2} + 4{b^2} + 8ab – 4{a^2} – 4{b^2} + 8ab – {b^3} + 4a – 12\)
\( = 4{a^2} – 4{a^2} + 4{b^2} – 4{b^2} + 8ab + 8ab – {b^3} + 4a – 12\)
\( = 16ab – {b^3} + 4a – 12\)
The value of expression for \(a = 3\) and \(b = 1\) is \(16(3)(1) – {1^3} + 4(3) – 12\)
\( = 48 – 1 + 12 – 12\)
\( =47 \)
Therefore, \(4\left( {{a^2} + {b^2} + 2ab} \right) – \left[ {4\left( {{a^2} + {b^2} – 2ab} \right) – \left\{ { – {b^3} + 4(a – 3)} \right\}} \right]\) when \(a = 3\) and \(b = 31\) is \(47\)

Summary

In this article, we have learnt the definition of algebraic expression, different types of algebraic expression, addition and subtraction of algebraic expression, use of brackets to solve the algebraic expression and orders of operations on algebraic expression. Also, we have solved some example problems on operations on algebraic expression.

FAQs

Q.1. How is the order of operations used when solving algebraic expressions?
Ans: The order of operations in algebra is important if you want to find the correct answer. We first run through any grouping symbols, then exponents or roots, multiplication, division, addition, and eventually subtraction. Solve expressions that need multiplication or division.

Q.2. How can we use the order of operations to evaluate algebraic expressions?
Ans: The order of operations to evaluate algebraic expressions follows the below steps:
1. We first need to solve the parentheses
2. Next, we need to solve the variables with exponents.
3. Multiplication and division are solved as they are encountered from left to right.
4. Addition and subtraction are solved as they are encountered from left to right.

Q.3. What are the four fundamental operations of algebraic expression?
Ans: As in arithmetic, the four fundamental operations of algebraic expressions are addition, subtraction, multiplication and division.

Q.4. What are algebraic expressions?
Ans: An algebraic expression is a combination of constants and variables connected by the signs of fundamental operations of addition, subtraction, multiplication and division.

Q.5. What is the correct order of operations in math?
Ans: The order of operations may be a rule that tells the right sequence of steps for evaluating a math expression. We will remember the order using PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Learn Operations on Algebraic Expressions

We hope this detailed article on the order of operations on algebraic expressions helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!