Algebraic Expressions: Definition, Formulas, Example

Algebraic Expressions: Algebraic Expressions are used to calculate solutions for any Mathematical operations that include variables such as addition, subtraction, multiplication or division. There are three types of algebraic expressions; Monomial Expression, Binomial Expression, and Polynomial Expression. Also, Boolean algebra plays a vital part in algebraic expressions.

An unknown value is represented as the letters x,y and z in the fundamentals of Algebraic expressions. These letters are referred to as variables. An algebraic expression can have both variables and constants. Together they form the algebraic expression. In this article, we will learn about algebraic expressions and Boolean algebra in detail.

Algebraic Expressions Definition for Boolean Algebra

Many students wonder what exactly are algebraic expressions or what is algebraic expressions definition. So, here is the answer: The combination of the constants and the variables connected by some or all of the four fundamental operations addition (( + ),) subtraction (( – ),) multiplication ((times)), and division (( ÷ )) is known as an algebraic expression.

Examples: (4x+5,10y – 5) are examples of the algebraic expression.

Algebraic Constant

The constant meaning in Maths is as follows:

Any quantity whose value never changes in the given subject (or boundary) of discussion is called a constant.

Example: (2,6,115,pi,e,……) are constants.

Algebraic Variable

The variable meaning in Maths is as follows:

Any quantity whose value changes in the given subject (or boundary) of discussion is called a variable.

Example: Conventionally (x,y,z…) are used as variables.

Note: Usually the small letters (x,y,z…) are used as variables and (a,b,c,p,q,r,…..) are used as constants. But, at times the letters (a,b,c,p,q,r,…..) are also used as variables. Hence, it is advisable to mention which letter is used as a variable and which letter is used as a constant for clarity in some cases.

Terms of Algebraic Expressions by Boolean Algebra

In an algebraic expression, a term may consist of \((i)\) only constant, \((ii)\) only one variable, \((iii)\) product of two or more variables,\((iv)\) a product of both the variable \((s)\) and the constant part. The terms may be positive or negative.

Example: (4, 17, x, y, xy, yz, 5xyz, 12xy, -4, -17, -x, -y, -xy, – yz, -5xyz, -12xy,…) are terms.

Coefficients

The fixed (or constant) number part along with the sign (positive or negative) associated with each algebraic term is called its coefficient.

Example: In the term (12xyz) the coefficient is (12).

In the term (-yz) the coefficient is (-1).

In the term (17) the coefficient is (17).

In the term (-4), the coefficient is (-4).

Degree

The degree of the polynomial is the highest integral power of the variable(s) of its terms when the polynomial is expressed in its standard form. It is the sum of exponents of the variables in the term if has more than one variable.

Examples: ({x^3}y + {x^2} + y + 13)
({x^3}y) has the degree (4) ((3) for (x) and (1) for (y))
({x^2}) has the degree (2)
(y) has the degree (1)
The constant term (17) has degree (0).
So, the highest degree among the three terms is (4) Hence, the polynomial is said to have a degree of (4).

Algebraic Expressions and Equations

An algebraic expression may contain one or more than one terms.

(i). If it contains one term, then it may be only one constant term or one term consisting of constants and variable.

Example: Each of (4, 17,- 4, -17, xy, -3yz,…)are Algebraic Expressions with only one term. 

(ii). If the Algebraic Expression contains more than one term then the different terms of the expression must be separated by either the (+) or (\) sign only.

Example: (4x + 5, 10y – 20, 3{x^2} + 2y – 5, xy+ x – y – 17,…)

Types of Algebraic Expressions

Below we have provided the 3 main types of algebraic expressions:

(i) Based on the number of the terms they contain
(ii) Based on the highest degree of the terms
(iii) Based on the number of variables it contains.

Algebraic Expressions and Equations Based on Number of Terms

The various algebraic expressions and equations based on a number of terms are as follows:

1. Monomial: It is an algebraic expression that contains only one term.
Example: (5x, 2xy, – 3{a^2}b, – 7), etc., are monomials.
2. Binomial: An algebraic expression that contains two terms is known as a binomial.
Example: ((2a+3b),(8 – 3x),({x^2} – 4x{y^2})), etc., are binomials.
3. Trinomial: An algebraic expression containing three terms is known as trinomial.
Example: ((a, + 2b + 5c), (x + 2y – 3z), ({x^3} – {y^3} – {z^3})) etc., are trinomials.
4. Quadrinomial: An algebraic expression containing four terms is known as a quadrinomial.
Example: ((x + y + z – 5), ({x^3} + {y^3} + {z^3} + 3xyz)) etc., are quadrinomials.
5. Polynomials: A expression containing two or more terms is known as a polynomial. It includes Binomials, Trinomials, Quadrinomials, and all the Algebraic expressions with five or more terms.

Algebraic Expressions Based on Highest Degree of Terms

The various algebraic expressions based on the highest degree of terms are as follows:

1. First Degree: It is an algebraic expression with degree (1).
Example: (5x,x,y,…)etc.

2. Second Degree: It is an algebraic expression with degree (2).
Example: (5{x^2}, {x^2} + 3xy + 12{y^2} + 3x – 8y + 9,…)etc.

3. Third Degree: It is an algebraic expression with degree (3).
Example: (5{x^3},{x^3} + 3xy + 12{y^2}, {y^2} + 3x – 8{y^3} + 9…)etc.

And so on.

Algebraic Expressions Based on Number of Variables It Contains

The various algebraic expression based on the number of variables it contains is as follows:

1. With One Variable: It is an algebraic expression with one variable only.
Example: (5x, x+2, y – 9,…), etc.

2. With Two Variables: It is an algebraic expression with two variables only.
Example: (7xy, 5{x^2}+z, {x^2}+3xy + 12{y^2}, {y^2}+3x – 8y + 9,…), etc.

3. With Three Variables: It is an algebraic expression with three variables only.
Example: (6xyz,5{x^3} + 3y + z, {x^3} + 3xy + 12{y^2}z,{y^2} + 3xz – 8{y^3} + 9,…)

And so on.

Like and Unlike Terms

Students can learn about like and unlike terms in an Algebraic expression worksheet below:

(i) Like Term: The terms having the same algebraic factors are known as like terms.

(ii) Unlike Term: The terms having different algebraic factors are known as, unlike terms.

Example: In the expression (2xy – 3x + 5xy – 4,) The terms (2xy) and (5xy) are like terms because they have the same algebraic factors (xy) but the terms (2xy) and (-3x) are unlike terms because they have the different algebraic factors (xy) and (x), respectively.

Arithmetic Operations on Algebraic Expressions

The arithmetic operations on algebraic expressions worksheets are given below:

Addition of Algebraic Expressions

In addition to algebraic expression, like terms are added with like terms only. Coefficients of the like terms are added. Unlike terms, if any will be left connected with the result with the mathematical operator it has.

Example: (3x + 5y + z + 7 + 4x + 9y + 11)/( = 3x + 4x + 5y + 9y + z + 7 + 11) write the like terms together, there is no other like term for (z) so, (= 7x, + 14y + z + 18) add all the pairs of like terms and connect the respective results with their respective mathematical signs.

Subtraction of Algebraic Expressions

To subtract an algebraic expression from another, change the signs (from(+{rm{to} {rm{ – }} ,{rm{or}}, {rm{from}}, {rm{ – to}} ,{rm{ + }}))of all the terms of the expressions which is to be subtracted and then the two expressions are added following the rules for addition.

Example: Subtract (( – 2{y^2} + frac{1}{2}y – 3)from,7{y^2} – 2y + 10)

It is done as follows: ((7{y^2} – 2y + 10) – left( { – 2{y^2} + frac{1}{2}y + 3} right))
= 7{y^2} – 2y + 10 + 2{y^2} – frac{1}{2}y + 3)
= 7{y^2} + 2{y^2} – 2y – frac{1}{2}y + 10 + 3) (Grouping like terms)
= (7 + 2){y^2} + left( { – 2 – frac{1}{2}} right)y + 13)
= 9{y^2} – frac{5}{2}y + 13)

Multiplication of Algebraic Expressions

General rule:

1. The product of two factors with the like signs are positive and the product of two factors with unlike signs is negative.
   i.e., (( + ) times ( + ) =  + )
   (( + ) times ( – ) =  – )
   (( – ) times ( + ) =  – )
   and (( – ) \times ( – ) =  + )
2. If (a) is any variable and (m,n) are positive integers then ({a^m} times {a^n} = {a^{m + n}})
   Example: ({a^3},times{a^5} = {a^{3 + 5}} = {a^8},{y^4}times y = {y^{4 + 1}} = {y^5})etc.

We have the following cases for multiplication:

Multiplication of Two Monomials

In this case, multiply the coefficient with the coefficient and the variable part with the variable part of the two monomials.
Example: (3ab times 5b = (3 times 5) times (ab times b) = 15a{b^2})

Multiplication of Monomial and a Binomial

In this case, follow the distributive law of multiplication over addition (a times (b + c) = a times b + a times c)

Example: (3x(4{x^2} + y + 2z) = 3x times 4{x^2} + 3x times y + 3x times 2z = 12{x^3} + 3xy + 6zx)

Multiplication of Two Binomials

In this case we follow the rule \((a + b) \times (c + d) = a \times (c + d) + b \times (c + d)\) and then follow the rule of multiplying a binomial with a monomial.

Example:
((3x + 2)(4{x^2} + y) = 3x times (4{x^2} + y) + 2 times (4{x^2} + y))
( = 3x times 4{x^2} + 3x times y + 2 times 4{x^2} + 2 \times y)
( = 12{x^3} + 3xy + 8{x^2} + 2y)
( = 12{x^3} + 8{x^2} + 3xy + 2y)

Division of Algebraic Expressions

Division of Algebraic identities can be done in two ways:

(i) Using Algebraic Identity

We can use any standard algebraic identity for division, as shown below:

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

Here, the algebraic identity ({a^3} – {b^3} = (a – b)({a^2} + ab + {b^2})) is used.

(ii) Using Long Division Method

In cases where algebraic identities cannot be used, the long division method is used in the same way we use it for the division of large numbers.

In this division, the dividend is ({x^2} + 3x + 1) , the divisor (x – 1) the quotient is (x + 2) and the remainder is (3)

Algebraic Expressions Formula

The algebraic expressions formula is the standard algebraic identities used to solve problems related to the algebraic expressions. We give below some of the formulae (the list is not exhaustive):

1. ({a^2} – {b^2} = (a – b)(a = b))
2. ({(a + b)^2} = {a^2} + 2ab + {b^2})
3. ({(a – b)^2} = {a^2} – 2ab + {b^2})
4. ({(a + b + c)^3} = {a^2} + {b^2} + {c^2} = 2ab + 2bc + 2ca)
5. ({(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} = {a^3} + {b^3} + 3ab(a + b))
6. ({(a – b)^3} = {a^3} – 3{a^2}b + 3a{b^2} – {b^3} = {a^3} – {b^3} – 3ab(a-b))
7. ({a^3} – {b^3} = (a-b)({a^2} + ab + {b^2}))
8. ({a^3} – {b^3} = (a-b)({a^2} + ab + {b^2}))
9. ({a^3} + {b^3} = (a + b)({a^2} – ab + {b^2}))
10. ({a^3} + {b^3} + {c^3} – 3abc = (a + b + c)({a^2} + {b^2} + {c^2} – ab – bc – ca))

Solved Examples of Algebraic Expressions

Students can check the solved examples on algebraic expressions below to prepare for various exams:

Q1. Add (5{x^2} – 7x + 3,) ( – 8{x^2} + 2x – 5) and (7{x^2} – x – 2)
A1. The solution is given below:
= (5{x^2} – 7x + 3) + ( – 8{x^2} + 2x – 5) + (7{x^2} – x – 2))
= 5{x^2} – 8{x^2} + 7{x^2} – 7x + 2x – x + 3 – 5 – 2) (collecting like terms)
= (5 – 8 + 7){x^2} + ( – 7 + 2 – 1)x + (3 – 5 – 2)) (adding like terms)
= 4{x^2} – 6x – 4).

Q2. Subtract ((2{x^2} – 5x + 7) from ((3{x^2} + 4x – 6))
A2. ((3{x^2} + 4x – 6) – (2{x^2} – 5x + 7))
= 3{x^2} + 4x – 6 – 2{x^2} + 5x – 7)
= {x^2} + 9x – 13).

Q3. Multiply: \( – 8a{b^2}c,3{a^2}b\) and \( – \frac{1}{6}\)
A3. (left( { – 8  times 3  times  frac{{ – 1}}{6}}  right)  times (3{a^2}b)  times  left( { –  frac{1}{6}}  right) ) ( left( { – 8  times 3  times  frac{{ – 1}}{6}}  right)  times (a  times {a^2}  times {b^2}  times b  times c) )  ( = 4{a^{(1 + 2)}} ,  times {b^{(2 + 1)}}  times c = 4{a^3}{b^3}c ).

Q4. Simplify the expression: \(12{m^2} – 9m + 5m – 4{m^2} – 7m + 10\).
A4. Rearranging the terms, we have:
= (12 – 4){m^2} + (5 – 9 – 7)m + 10)
= 8{m^2} + ( -4 -, -7)m + 10)
= 8{m^2} + ( – 11)m + 10)
= 8{m^2} – 11m + 10).

Q5. What is the degree of the monomial (7)?
A5. We know that degree of any constant term is zero ((0)) Hence, the degree of the monomial (7) is (0).

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FAQs on Algebraic Expressions

Q.1: What is algebraic expression and equation?
Ans: The algebraic expression is a combination of numbers and variables and operation symbols. An equation is made up of two expressions linked by an equal sign.

Q.2: What are basic algebraic expressions?
Ans: The combination of the constant and the variables that are linked by the signs of fundamental operations of addition, subtraction, multiplication, and division is called an algebraic expression.

Q.3: What are the types of algebraic expressions?
Ans: In terms of the number of terms present, the algebraic expressions are classified as Monomial, Binomial, Trinomial, Quadrinomial, Polynomial, etc.
In terms of the number of variables present, the algebraic expressions are classified as expressions with one variable, two variables, three variables, etc.

Q.4: What is an algebra formula?
Ans: The different formulae used in solving problems related to Algebraic expressions called algebra formula. Some of them are:
({a^2} – {b^2} = (a – b)(a + b))
({(a + b)^2} = {a^2} + 2ab + {b^2})
({(a – b)^2} = {a^2} – 2ab + {b^2})
({(a + b + c)^3} = {a^2} + {b^2} + {c^2} = 2ab + 2bc + 2ca)
({(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} = {a^3} + {b^3} + 3ab(a+b))
({(a – b)^3} = {a^3} – 3{a^2}b + 3a{b^2} – {b^3} = {a^3} – {b^3} – 3ab(a – b))
({a^3} – {b^3} = (a-b)({a^2} + ab + {b^2}))
({a^3} + {b^3} = (a + b)({a^2} – ab + {b^2}))
({a^3} + {b^3} + {c^3} – 3abc = (a + b + c)({a^2} + {b^2} + {c^2} – ab – bc – ca))

Q.5: Is 5 an algebraic expression?
Ans: Yes, 5 is an algebraic expression. It is a monomial having only one constant term (5).