What are the types of algebraic expressions?

An algebraic expression is an expression composed of various components, such as the variables, constants, coefficients, and arithmetic operations. These components form various parts of the algebraic expressions. An algebraic expression is a linear equation composed of any number of variables. The highest power of the variable is known as the degree. An algebraic expression containing one variable is monomial, two variables is binomial, and so on. For instance, if we assume an expression to be, 2x+4y-9  

An algebraic expression is formed by various components known as terms. The terms may be coefficients, variables, and their combination, or constants. 

Types of Algebraic Expressions

The algebraic expressions are divided into various parts depending on the variables that are found in the corresponding algebraic expression. It is also affected by the following factors, them being, the total number of terms forming the algebraic expression and the values of the exponents of the variables in each expression.

Monomial

An algebraic expression comprising only one term, where all of the exponents of all the variables are non-negative integers. A monomial can be used to depict a quantity in algebraic form. Basically, a polynomial with one term is known as a monomial. The monomials can be expressed mathematically in the following two different algebraic forms.

1. Algebraic terms
   Every algebraic term is an algebraic expression in a single term. 
   Example- x, y, ab, 2x²y, 
2. Numbers
   Every number is considered to be a monomial.
   Example- 2, 5, -9,  , 

Examples of Monomial

• x is a monomial in one variable x.
• 28mn² is a monomial in two variables m and n
• -7mno is a monomial in three variables m, n, and o.

Binomial

An algebraic expression comprising only two terms, where all of the exponents of all the variables are non-negative integers. A binomial can be used to depict two unlike algebraic terms showing a quantity, which are connected using the elementary mathematical addition and subtraction operators. Basically, a polynomial with two terms is known as a binomial. It is possibly formed in two different forms.

1. A term and a number
   One term is an algebraic term and another term may be a number.
   Example- 12a + 18, a – 28,   + ab²
2. Two Unlike Algebraic terms
   Two, unlike algebraic terms, may form a binomial.
   Example- x + y, 

Examples of Binomial

• x + y is a binomial in two variables x and y.
• m² + 2n is a binomial in two variables m and n.
• -14p – q² is a binomial in two variables p and q.

Trinomial

An algebraic expression comprising three terms, where all of the exponents of all the variables are non-negative integers. A trinomial can be used to depict three unlike algebraic terms showing a quantity, which are connected using the elementary mathematical addition and subtraction operators. Basically, a polynomial with three terms is known as a trinomial. It is possibly formed in two different forms.

1. Three Unlike Algebraic Terms
   Three, unlike algebraic terms, may form a trinomial. 
   Example- a – b + c, 2mn + m²n – m³n³
2. Two terms and a Number
   Two terms are an algebraic term and another term may be a number. 
   Example- x + y + 71,  mn² – mn – 78

Examples of Trinomial

• a + b + c is a trinomial in three variables namely a, b and c.
• mn + m + 27n² is a trinomial in two variables m and n.
• -47x⁴ + y³ – 14x²y² is a trinomial in two variables x and y.

Polynomial

An algebraic expression comprising more than three terms, where all of the exponents of all the variables are non-negative integers. A polynomial can be used to depict three unlike algebraic terms showing a quantity, which are connected using the elementary mathematical addition and subtraction operators. Basically, a polynomial contains more than three terms. It is possibly formed in two different forms.

1. One or More term
   Example- -18x, x + y + z, 45x² + 12y² – z²
2. A number with terms
   Example- 45, x – 13, m + mn + 13m²n

Example of Polynomial

• 42x + 15y is a polynomial of two terms in two variables x and y.
• 49ab + 55a + 25 is a polynomial of three terms in two variables a and b.

Multinomial

An algebraic expression comprising more than three terms, where all of the exponents of all the variables are either positive or negative integers. A polynomial can be used to depict three unlike algebraic terms showing a quantity, which are connected using the elementary mathematical addition and subtraction operators It is possibly formed in two different forms.

1. Greater than One Term
   Two or more, unlike algebraic terms.
   Example- x + 5y, a + 14ab – c
2. One or more terms and a Number
   A number and one or more algebraic terms.
   Example- y – 15, x ²+ y² – 0.15

Examples of Multinomials

• x + y is a multinomial of two terms in two variables x and y.
• p + q + r + s is a multinomial of four terms in four variables p, q, r, and s.

Sample Questions

Question 1. There are 25 oranges in a bag. Form the algebraic expression depicting the number of oranges in terms of x number of bags.

Solution:

Since, we have, 

The number of oranges in one bag = 25. 

Here, 

The number of bags = x. So,

the number of oranges in x bags = 25x. 

Therefore,

Required Algebraic Expression = 25x

Question 2. What type of algebraic expression is 4x + 5?

Solution:

4x + 5 has two monomials 4x and 5 and hence it is a binomial. Now, since every binomial is a type of polynomial. 

Therefore,

4x + 5 is a polynomial/binomial.

Question 3. Define the possible way to derive algebraic expressions?

Solution:

An algebraic expression is considered to be a combination of constants, variables, and algebraic operations (+, -, ×, ÷). 

For example, let us assume Yash age is thrice more than Mallika. Also, the total age of Yash and Mallika is equivalent to 40. Deriving an algebraic expression for this, we have, 

3x + x = 40 ⇒ 4x = 40; where x is the age of Mallika.

Question 4. Simplify the expressions by joining the like terms and also find which algebraic expression is formed?

• 2xy³ + 3x²y³ + 6y³x
• 15m³ – 2m + 5m + 12m³ – 13m + 10m – 14m³

Solution:

Here is the table below of the solution:

S.no	Term	Simplify	Algebraic Expression
1	2xy3 + 3x2y3 + 6y3x	3x2y3 + 8xy3	Binomial
3	15m3 – 2m + 5m + 12m3 – 13m + 10m – 14m3	13m3	Monomial