Operations on Algebraic Expressions: Types, Methods

Operations on Algebraic Expressions: Algebraic expressions are the type of expressions that mainly consists of three parts: variables, constants and coefficients. The four basic operations are addition, subtraction, multiplication, and division. Operations on algebraic expressions are the process through which we can solve complex as well as simple equations.

The algebraic expressions are divided into 3 types namely, monomial, binomial, and polynomial or multi-term expressions. In this article, we are going to study the operations of algebraic expressions in detail. Continue reading to learn more and for answers to common search queries such as division of algebraic, expressions, multiplication of algebraic expressions, subtraction of algebraic expressions and so on.

What is an Algebraic Expression?

An algebraic expression is a combination of terms connected by the basic mathematical symbols \(+\) or \(-.\) For example, let us consider the expression \(8x + 7\)

In this algebraic expression, two terms \(8x\) (variable term) and \(7\) (constant term) are connected by \(+\) sign. 

Algebraic Terms 

The algebraic terms are as follows:

• Variables: Alphabetic letters along with integers/fractions or alphabetic letters alone.
• Coefficients: The numbers that are associated with the variables in a single term.
• Constants: Numbers or integers alone that are usually associated with other terms by basic operations.
• Examples: \(8 x y z, 25 x+12 y+9,2 y z^{2}-3 z y, 3 a+2 b+5 c\), etc.

Types of Algebraic Expression

The algebraic expressions are categorised into three types. They are:

1. Monomial or single term expressions. For example \(4 x y^{2}, 3 a b, 7 p, 5 x y z\), etc., where \(x, y, z, a, b, p\) are the variables and \(3, 4, 5, 7\) are the coefficients.
2. Binomial or two terms expressions. For example, \(2 x y-x, p q-5 p^{2}\) etc.
3. Polynomial or multi-term expressions. For example, \(2 x+5 y-4,2 x y^{2}+3 y+1\), etc.

Order of Operations on Algebraic Expressions

Operations on algebraic expressions are worked out by following a certain order. We break down any given algebraic expression into order and solve it. Use the acronym BEDMAS to recall the order of operation.

1. \(B\) stands for Brackets: Any parenthesis, if present, needs to be first solved.
2. \(E\) stands for Exponents: Exponents are operated next to brackets.
3. \(D\) stands for Division: Check if there are any divisions to be done.  
4. \(M\) stands for Multiplication: The fourth operation in the order is multiplications.
5. \(A\) stands for Addition: Next, you need to do the addition.
6. \(S\) stands for Subtraction: The last operation will be subtraction.

Fundamental Operations on Algebraic Expressions

The mathematical operations that are performed on numbers are also performed on algebraic expressions. The addition and subtraction of algebraic expressions are very similar to the addition and subtraction of numbers. In the case of algebraic expressions, we group the like terms and unlike terms together and simplify.

The multiplication of algebraic expression involves properties like the distributive and commutative properties of addition. That will be convenient while multiplying polynomials. Division of algebraic expressions is performed in the same way as dividing two whole numbers or fractions. Division of two algebraic expressions or variable expressions involves taking out common terms and cancelling them out. These common terms either include constants, variables, terms, or coefficients alone.

We will discuss the four operations of algebraic expression in this section

1. Addition of algebraic expressions
2. Subtraction of algebraic expression
3. Multiplication of algebraic expression 
4. Division of algebraic expression

Addition of Algebraic Expressions

To add the algebraic expressions, follow the steps below:

Step 1: Sort out all the like terms based on the variables, i.e., the term consisting of the same variables are grouped.

Step 2: Addition is performed on all the grouped terms with the same variables by adding the coefficients and should be written with a single coefficient term.

Step 3: Similarly, for all the like terms, the operations are to be done.

Step 4: For constants, add them like the usual addition of numbers

Step 5: If there are no like terms present, then keep the expression as it is.

There are two methods to add algebraic expressions.

1. Horizontal method
2. Column method

1. Horizontal Method 

Let us add these expressions with the help of the horizontal method: \((xy+2yz+4), (2yz+4xy+6 ).\)

\((xy+2yz+4)+(2yz+4xy+6 )\)

To add, we first open the brackets and group the like terms

Open the brackets: \(xy+2yz+4+2yz+4xy+6=xy+4xy+2yz+2yz+4+6\)

Add the coefficient of like terms to simplify: \(5xy+4yz+10.\)

2. Column Method 

The expressions are written column-wise in the column method one below the other, taking care that like terms are written one below the other. Then, we add the numerical coefficient of each column (like terms) and write the sum below along with the variable. Let’s take the above example to add using the column method.

Subtraction of Algebraic Expressions

The subtraction of algebraic expressions is similar to the addition. While subtracting one algebraic expression from another, we have to be careful with the signs. Suppose there is a subtraction sign before the bracket. When the bracket is opened, we need to reverse the signs of all terms inside the bracket. It is recommended to use the column subtraction method. There are two methods to subtract algebraic expressions, namely, horizontal and column methods.

1. Horizontal Method

Let’s solve: \((5x+2y–7z)-(4x-4y+9z+5)\) using the horizontal method.

First, open the brackets as follow:

\((5 x+2 y-7 z)-(4 x-4 y+9 z+5) \Rightarrow 5 x+2 y-7 z-4 x+4 y-9 z-5\)

Group the like terms and simplify the expression: 

\(=5x-4x+2y+4y-7z-9z-5\) 

\(=x+6y-16z-5\)

2. Column Method

Let us subtract the above expressions using the column method. Here we place the two expressions, one below the other, we change the signs of all terms in the second expression as shown below, and then we simplify the expressions.

Multiplication of Algebraic Expressions

In the multiplication of algebraic expressions, let us look at two simple rules.

(i) The product of two terms with like signs is positive. The product of two terms with unlike signs is negative.
(ii) If \(x\) is a variable and \(m, n\) are positive integers, then \(\left(x^{m} \times x^{n}\right)=x^{m+n}\)

Follow the steps below for the multiplication of algebraic expressions

Step 1: The multiplication is done by multiplying each term of the first expression with each term of the second expression.

Step 2: Add the powers and express as exponent with the variable if the same variables appear.

Step 3: If different variables exist, just write them as the product of another variable.

Step 4: Every term obtained after multiplication should be separated by its respective signs.

I. Multiplication of Two Monomials

Rule:

Product of two monomials = (product of their numerical coefficients) \(×\) (product of their variables)

Example: \(6 x^{2} y^{2}\) and \(-3 x^{2} y^{3}\)

\(\left( {6{x^2}{y^2}} \right) \times \left( { – 3{x^2}{y^3}} \right) = \{ 6 \times ( – 3)\}  \times \left\{ {{x^2}{y^2} \times {x^2}{y^3}} \right\}\)

\(=-18 x^{2+2} y^{2+3}\)

\(=-18 x^{4} y^{5}\)

II. Multiplication of a Polynomial by a Monomial

Rule:

Multiply all the terms of the polynomial by the monomial, using the distributive law:

\(a \times(b+c)=a \times b+a \times c\)

Example: \(5 a^{2} b^{2} \times\left(3 a^{2}-4 a b+6 b^{2}\right)\)

\(5 a^{2} b^{2} \times\left(3 a^{2}-4 a b+6 b^{2}\right)=\left(5 a^{2} b^{2}\right) \times\left(3 a^{2}\right)+\left(5 a^{2} b^{2}\right) \times(-4 a b)+\left(5 a^{2} b^{2}\right) \times\left(6 b^{2}\right)\)

\(=15 a^{4} b^{2}-20 a^{3} b^{3}+30 a^{2} b^{4}\)

III. Multiplication of Two Binomials

Suppose \((a+b)\) and \((c+d)\) are two binomials. The product may be found as given below.

\((a+b) \times(c+d)=a \times(c+d)+b \times(c+d)=(a \times c+a \times d)+(b \times c+b \times d)\)
\(=a c+a d+b c+b d\)

Example: Multiply \((3x+5y)\) and \((5x-7y).\)

\((3x+5y)×(5x-7y)=3x×(5x-7y)+5y×(5x-7y) =(3x×5x-3x×7y)\)
\( + \left( {5y \times 5x – 5y \times 7y} \right) = \left( {15{x^2} – 21xy} \right) + \left( {25xy – 35{y^2}} \right) = 15{x^2} – 21xy + 25xy – 35{y^2}\)
\( = 15{x^2} + 4xy – 35{y^2}\)

IV. Multiplication of Polynomials

Let us multiply two polynomials 

Example: \(\left(2 x^{3}-5 x^{2}-x+7\right) \times\left(3-2 x+4 x^{2}\right)\)

We arrange the terms of the given polynomials in decreasing power of \(x\) and then multiply,

\(\left(2 x^{3}-5 x^{2}-x+7\right) \times\left(3-2 x+4 x^{2}\right)=\left(2 x^{3}-5 x^{2}-x+7\right) \times\left(4 x^{2}-2 x+3\right)\)

\(=8 x^{5}-20 x^{4}-4 x^{3}+28 x^{2}-4 x^{4}+10 x^{3}+2 x^{2}-14 x+6 x^{3}-15 x^{2}-3 x+21\)

\(=8 x^{5}-24 x^{4}+12 x^{3}+15 x^{2}-17 x+21\)

Division of Algebraic Expressions

In dividing an algebraic expression we take out the common terms and cancel them, similar to the division of the numbers. Here, the common terms correspond to either constants, variables, terms, or just coefficients

There are different types of division of algebraic expressions. The different types of division of algebraic expressions are as follows:

1. Division of monomial by a monomial
2. Division of polynomial by a monomial
3. Division of polynomial by a polynomial

1. Division of Monomial by a Monomial

A monomial is a type of expression that has only one term. The method to perform the division of a monomial by a monomial is explained below with an example.

Example: \(27 x^{4} \div 3 x\)

Here 3x and \(27 x^{4}\) be the two monomials.

\(27 x^{4} \div 3 x=\frac{27 \times x \times x \times x \times x}{3 \times x}\) [Writing both the terms by doing their prime factorisation]

Now, cancel the common term, which is \(3x.\)

Thus, \(27 x^{4} \div 3 x=9 x^{3}\)

2. Division of Polynomial by a Monomial

A polynomial is an expression that may be binomial, trinomial or an equation with \(n\)-terms.

Example: \(\left(4 y^{5}+5 y^{3}+6 y\right) \div 2 y\)

Here, the polynomial is \(4 y^{5}+5 y^{3}+6 y\) and the monomial is \(2y.\)

Taking the common factor \(2y,\) it becomes: \(4 y^{5}+5 y^{3}+6 y=2 y\left(2 y^{4}+\frac{5}{2} y^{2}+3\right)\)

The common terms of the numerator and denominator may be cancelled

\(\Rightarrow \frac{2 y\left(2 y^{4}+\frac{4}{2} y^{2}+3\right)}{2 y}=\left(2 y^{4}+\frac{5}{2} y^{2}+3\right)\)

Thus, \(\left(4 y^{5}+5 y^{3}+6 y\right) \div 2 y=2 y^{4}+\frac{5}{2} y^{2}+3\)

3. Division of Polynomial by a Polynomial

Let us consider an example and divide the polynomial by a polynomial. Here also we focus on common terms which can be cancelled from both numerator and denominator.

Example: \(\left(7 y^{2}+14 y\right) \div(y+2)\)

Here, both polynomials are in binomial form.

Take out the common term : \(\left(7 y^{2}+14 y\right)=7 y(y+2)\)

\(\Rightarrow \frac{\left(7 y^{2}+14 y\right)}{(y+2)}=\frac{7 y(y+2)}{(y+2)}\)

Eliminate \((y+2)\) from the numerator and denominator, we get,

\(\Rightarrow \frac{7 y(y+2)}{(y+2)}=7 y\)

Thus, \(\left(7 y^{2}+14 y\right) \div(y+2)=7 y\)

Division of Algebraic Expression Using Algebraic Identities

Division of algebraic expression can be done using standard algebraic identities such as

1. \(a^{2}-b^{2}=(a-b)(a+b)\)
2. \((a+b)^{2}=a^{2}+2 a b+b^{2}\)
3. \((a-b)^{2}=a^{2}-2 a b+b^{2}\)
4. \((a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a\)
5. \((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}=a^{3}+b^{3}+3 a b(a+b)\)
6. \((a-b)^{3}=a^{3}-3 a^{2} b+3 a b^{2}-b^{3}=a^{3}-b^{3}-3 a b(a-b)\)
7. \(a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\)
8. \(a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)\)
9. \(a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)\)

Example: We can divide \(x^{3}-8\) by \(x-2\) as follows:

\(\frac{x^{3}-8}{x-2}=\frac{(x)^{3}-(2)^{3}}{x-2}=\frac{(x-2)\left[(x)^{2}+x \times 2+(2)^{2}\right]}{x-2}=\frac{(x-2)\left[x^{2}+2 x+4\right]}{(x-2)}\)

\(=x^{2}+2 x+4\)

Here, the algebraic identity \(a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\) is used.

Division of Algebraic Expression Using Long Division Method

Example:

Solved Examples of Operations on Algebraic Expressions:

Q.1. Add 5x+6y+7z and 6x-7y+3
Ans: Given, \(5x+6y+7z\) and \(6x-7y+3\)
Now let us group the like terms to add the given expressions
\(5x+6y+7z+6x-7y+3=5x+6x+6y-7y+7z+3\)  (adding like terms by adding their coefficients)
\(=11x-y+7z+3\)
Thus, \(5x+6y+7z+6x-7y+3=11x-y+7z+3.\)

Q.2.  Subtract (8y-7x-1) from (5x-7y+9) 
Ans: \((5 x-7 y+9)-(8 y-7 x-1)=5 x-7 y+9-8 y+7 x+1\) (Signs of the terms inside the bracket changes on opening the bracket)
\(=5x+7x-7y-8y+9+1\) (grouping the like terms)
\( = 12x – 15y + 10\)
Thus, \((5 x-7 y+9)-(8 y-7 x-1)=12 x-15 y+10\)

Q.3. Multiply 5x+6y and 3x+2y
Ans: Each term of the first expression must be multiplied by each term in the second expression
\((5 x+6 y)(3 x+2 y)=5 x(3 x+2 y)+6 y(3 x+2 y)\)
\(=15 x^{2}+10 x y+18 x y+12 y^{2}\)
\(=15 x^{2}+28 x y+12 y^{2}\)
Thus, \((5 x+6 y)(3 x+2 y)=15 x^{2}+28 x y+12 y^{2}\)

 Q.4. Simplify: \(\left(4 x^{2} y-6 x y^{2}+4 x y\right) \div 2 x y\)
Ans: To divide \(\left(4 x^{2} y-6 x y^{2}+4 x y\right) \div 2 x y\)
\(\frac{\left(4 x^{2} y-6 x y^{2}+4 x y\right)}{2 x y}=\frac{2 x y(2 x-3 y+2)}{2 x y}\)
The common terms in the numerator and denominator are cancelled
\(=(2 x-3 y+2)\)
Thus, \(\left(4 x^{2} y-6 x y^{2}+4 x y\right) \div 2 x y=(2 x-3 y+2)\)

Q.5. Simplify: \(7 a b^{2} \times\left(-4 a^{2} b c\right) \div(-14 a b c)\)
Ans: \(7 a b^{2} \times\left(-4 a^{2} b c\right) \div(-14 a b c)=\frac{7 a b^{2} \times\left(-4 a^{2} b c\right)}{-14 a b c}\)
\(=\frac{-28 a^{3} b^{3} c}{-14 a b c}\)
\(=2 a^{2} b^{2}\)
Thus, \(7 a b^{2} \times\left(-4 a^{2} b\right) \div(-14 a b c)=2 a^{2} b^{2}\)

Summary

In this article, we discussed the basic operations of algebraic expression. The algebraic expressions are used to solve mathematical equations. There are 3 types of algebraic expressions namely, monomial, binomial, and polynomial expressions. The basic algebraic operations are addition, subtraction, multiplication, and division. The division of algebraic expressions is divided into 3 types namely, division of monomial by a monomial, division of a polynomial by a monomial, and division of a polynomial by a polynomial.

FAQs on Operations of Algebraic Expressions

Q.1: What are the basic algebraic operations?
Ans: The basic algebraic operations are addition, subtraction, multiplication, and division.

Q.2: How do you solve algebraic operations?
Ans: The order of operations is as follows: 
1) Simplify terms inside parentheses or brackets
2) Simplify exponents and roots
3) Perform multiplication and division
4) Perform addition and subtraction

Q.3: What is an example of algebraic expression?
Ans: \(2 x, 3 x y^{2}, 9 a c+3 b c, 2 p q+3 p^{3}+4 q\) are some of the examples of an algebraic expression.

Q.4: What are the types of algebraic expressions?
Ans: There are three types of algebraic expression:
1. Monomial Expression.
2. Binomial Expression.
3. Polynomial Expression.

Q.5: What are monomials and polynomials?
Ans: Monomial is a polynomial with one term. In comparison, a polynomial is an expression that contains variables and coefficients irrespective of the number of terms.

We hope this article on the operation of algebraic expressions is helpful to you.