Algebraic expressions: definition, formulas, example (2023)

Algebraic Expressions: Algebraic expressions are used to calculate solutions to any mathematical operation involving variables such as addition, subtraction, multiplication or division. There are three types of algebraic expressions; monomial expression, binomial expression and polynomial expression. In addition, Boolean algebra plays an important role in algebraic expressions.

An unknown value is represented as the letters x, y, and z in the basics of algebraic expressions. These letters are called 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.

Definition of algebraic expressions for Boolean algebra

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

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

Algebraic constant

The constant meaning in mathematics is as follows:

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

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

Algebraic Variable

The meaning of variables in mathematics is as follows:

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

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

note:Usually the lower case letters (x,y,z...) are used as variables and (a,b,c,p,q,r,.....) as constants. But sometimes the letters (a,b,c,p,q,r,…..) are also used as variables. Therefore, it is advisable to mention in some cases which letter is used as a variable and which letter is used as a constant.

Terms of algebraic expressions by Boolean algebra

In an algebraic expression, an expression consisting 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 can be positive or negative.

Beispiel: (4, 17, x, y, xy, yz, 5xyz, 12xy, -4, -17, -x, -y, -xy, – yz, -5xyz, -12xy,…) sind Terme.

coefficients

The fixed (or constant) number part, together 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).

Grad

The degree of the polynomial is the highest integer power of the variables of its terms when the polynomial is expressed in its standard form. It is the sum of the exponents of the variables in the expression when there is more than one variable.

Examples: ({x^3}y + {x^2} + y + 13)
({x^3}y) has degree (4) ((3) for (x) and (1) for (y))
({x^2}) has degree (2)
(y) graduated (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 degree (4).

Algebraic expressions and equations

An algebraic expression can contain one or more terms.

(I). If it contains a term, then it can only be a constant term or a term made up of constants and variables.

Example: (4, 17,- 4, -17, xy, -3yz,...) are each algebraic expressions with only one term.

(ii). If the algebraic expression contains more than one term, the different terms of the expression need only be separated by the (+) or (\) sign.

Beispiel: (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 terms included
(ii) Based on the highest degree of terms
(iii) Based on the number of variables it contains.

Algebraic expressions and equations based on number of terms

The differentalgebraic expressions and equationsbased 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 containing two terms is called a binomial.
Example: ((2a+3b),(8 – 3x),({x^2} – 4x{y^2})) etc. are binomials.
3.Trinomial: An algebraic expression containing three terms is called a trinomial.
Example: ((a, + 2b + 5c), (x + 2y – 3z), ({x^3} – {y^3} – {z^3})) etc. are trinominals.
4.Quadrinome: An algebraic expression that contains four terms is called a quadrinomial.
Example: ((x + y + z – 5), ({x^3} + {y^3} + {z^3} + 3xyz)) etc. are quadrinominals.
5.Polynomials: An expression containing two or more terms is called a polynomial. It includes binomials, trinomials, quadrinomials, and all algebraic expressions with five or more terms.

Algebraic expressions based on the 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 of 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.

Etc.

Algebraic expressions based on the number of variables contained

The various algebraic expressions based on the number of variables included are as follows:

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

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

Etc.

Like and contrary to terms

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

(I)Same term: The terms with the same algebraic factors are called similar terms.

(ii)Unlike term: The terms with different algebraic factors are called unlike terms.

Example: In the expression (2xy – 3x + 5xy – 4,) the terms (2xy) and (5xy) are equal terms because they have the same algebraic factors (xy) but the terms are (2xy) and (-3x). not equal to terms because they have different algebraic factors (xy) and (x) respectively.

Arithmetic operations on algebraic expressions

The arithmetic operationsAlgebraic expression worksheetsare given below:

Adding algebraic expressions

In addition to the algebraic expression, equal terms are only added with equal terms. Coefficients of equal terms are added. Unlike terms, any terms remain associated with the result with whatever mathematical operator they have.

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

Subtraction of algebraic expressions

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

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

This 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) (group 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 same sign is positive and the product of two factors with different signs is negative.
   d.h. (( + ) mal ( + ) = + )
   (( + ) mal ( – ) = – )
   (( – ) mal ( + ) = – )
   and (( – ) \times ( – ) = + )
2. If (a) is any variable and (m,n) are positive integers then ({a^m} times {a^n} = {a^{m + n}})
   Beige: ({a^3},bad {a^5} = {a^{3 + 5}} = {a^8},{y^4}bad y = {y^{4 + 1}} = {y^5}) usw.

We have the following cases for multiplication:

Multiplication of two monomials

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

Multiplication of a monomial and a binomial

In this case, obey the distribution 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 wrong (4{x^2} + y) + 2 wrong (4{x^2} + y))
( = 3x mal 4{x^2} + 3x mal y + 2 mal 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

Splitting algebraic identities can be done in two ways:

(i) Use of 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 mal 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 the 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 dividing large numbers.

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

Formula for algebraic expressions

The formulas for algebraic expressions are the standard algebraic identities used to solve problems related to the algebraic expressions. We give some of the formulas below (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 review the solved algebraic expression examples below to prepare for various exams:

Q1. Addiere (5{x^2} – 7x + 3,) ( – 8{x^2} + 2x – 5) und (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) (collect similar terms)
= (5 – 8 + 7){x^2} + ( – 7 + 2 – 1)x + (3 – 5 – 2)) (add 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 the degree of every constant term is zero ((0)). Hence the degree of the monomial (7) is (0).

You can also check

FAQs on algebraic expressions

F.1: What is an algebraic expression and equation?
Answer: The algebraic expression is a combination of numbers and variables and operation symbols. An equation consists of two expressions joined by an equals sign.

F.2: What are basic algebraic expressions?
Answer: The combination of the constants and the variables connected by the signs of the basic operations of addition, subtraction, multiplication and division is called an algebraic expression.

Q.3: What types of algebraic expressions are there?
Answer: 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 one-variable, two-variable, three-variable, and so on.

Q.4: What is an algebra formula?
Answer: The various formulas used to solve problems related to algebraic expressions are called the algebraic 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))

F.5: Is 5 an algebraic expression?
Answer:Yes, 5 is an algebraic expression. It is a monomial with only one constant term (5).