Algebraic Expressions

Algebraic expression started in the 9th century. In the beginning, it was more in statement form and not mathematical at all. For instance, algebraic equations used to be written as “5 times the thing added with 3 gives 18, which is 5x + 3 = 18. This type of equation which was not mathematical was Babylonian algebra. Algebra evolved with time and with the different forms provided. It started with Egyptian algebra, then came Babylonian algebra, then came Greek geometrical algebra, moved to diophantine algebra, followed by Hindu algebra, then came Arabic algebra, and followed by abstract algebra. Today, the easiest and most convenient form of algebra is taught in classes for better understanding.

What are Algebraic Expressions?

Algebraic expressions are the expressions obtained from the combination of variables, constants, and mathematical operations like addition, subtraction, multiplication, division, and so on. An algebraic expression is made up of terms, there can be one or more than one terms present in the equation. Let’s learn about the basic terms used in algebraic expressions.

Algebraic Terms, Constants, Variables, and Coefficients

In the algebraic expression, fixed numerals are called constants. Constants do not have any variables attached to them. For example, 3x – 1 has a constant -1 to it. Variables are the unknown values that are present in the algebraic expression. For instance, 4y + 5z has y and z as variables. Coefficients are the fixed values (real numbers) attached to the variables. They are multiplied by the variables. For example, 5x² + 3 has 5 as the coefficient of x². A term can be a constant, a variable, or a combination of both. Each term is separated by either addition or subtraction. For example, 3x + 5, 3x, and 5 are the terms.

Types of Algebraic Expressions

There are various types of algebraic expressions based on the number of terms.

Type of Algebraic Expression	Definition	Examples
Monomial	A monomial algebraic expression is an algebraic expression that has only one term.	2xy, 5y3, 7a, 2b, etc
Binomial	A binomial algebraic expression is an algebraic expression that has two unlike terms or two monomials.	6x + 8y, 3a2 + 2ab, etc.
Trinomial	A trinomial algebraic expression is an algebraic expression that has three unlike terms or three monomials.	x + y + z, 11x2 – 5y + z, etc
Polynomial	A polynomial algebraic expression is an algebraic expression that has two or more terms with non-negative integral exponents of a variable.	3x + 4y + 5z, ax3 + bx2 + cx + d, etc
Multinomial	A multinomial algebraic expression is an algebraic expression that has one or more than one term. Here, the exponent of variable can be negative also.	x + y-2, 5x2 + 6z-1

Simplifying Algebraic Expressions

Simplifying algebraic expressions is easy and very basic. First, understand what are like and unlike terms. Like terms have the same sign and unlike terms have opposite signs. To simplify the given algebraic expression, first, find out the terms having the same power. Then, if the terms are like terms, add them; if they are unlike terms, find the difference between the terms. The most simplified form of an algebraic expression is the one where no same power terms are not repeated. 

For instance, lets simplify 4x⁵ + 3x³ – 8x² + 67 – 4x² + 6x³, the same powers that are repeated are cubic and square, upon combining them together, the expression becomes, 4x⁵ + (3x³ + 6x³) – (8x² – 4x²) + 67. Now, simplifying the expression, the final answer obtained is 4x⁵ + 9x³ – 12x² + 67. This term does not have any terms repeated that have the same power.

Addition of Algebraic Expressions

When an addition operation is performed on two algebraic expressions, like terms are added with like terms only, i.e., coefficients of the like terms are added.

Example: Add (25x + 34y + 14z) and (9x − 16y + 6z + 17).

Solution:

(25x + 34y + 14z) + (9x − 16y + 6z + 17)

By writing like terms together, we get

= (25x + 9x) + (34y − 16y) + (14z + 6z) + 17

By adding like terms, we get

= 34x + 18y + 20z + 17.

Hence, (25x + 34y + 14z) + (9x − 16y + 6z + 17) = 34x + 18y + 20z + 17.

Subtraction of Algebraic Expressions

To subtract an algebraic expression from another, we have to add the additive inverse of the second expression to the first expression.

Example: Subtract (5b² + 6b + 8) from (3b² − 5b).

Solution:

(5b² + 6b + 8) − (3b² − 5b)

= (5b² + 6b + 8) + (−3b² + 5b)

= (5b² − 3b²) + (6b + 5b) + 8 = 2b² + 11b + 8

Multiplication of Algebraic Expressions

When a multiplication operation is performed on two algebraic expressions, we have to multiply every term of the first expression with every term of the second expression and then combine all the products.

Example: Multiply (3x + 2y) with (4x + 6y − 8z)

Solution:

(3x + 2y)(4x + 6y − 8z) = 3x(4x) + 3x(6y) − 3x(8z) + 2y(4x) + 2y(6y) − 2y(8z)

= 12x² + 18xy − 24xz + 8xy + 12y² − 16yz

= 12x² + 12y² + 26xy − 16yz − 24xz

Division of Algebraic Expressions

When we have to divide an algebraic expression from another, we can factorize both the numerator and the denominator, then cancel all the possible terms, and simplify the rest, or we can use the long division method when we cannot factorize the algebraic expressions.

Example: Solve: (x² + 5x + 6)/(x + 2)

Solution:

= (x² + 5x + 6)/(x + 2)

(x² + 5x + 6) = (x + 2) (x + 3)

= [(x + 2) (x + 3)]/(x + 2)

= (x + 3)

Algebraic Formulas

The general algebraic formulas we use for solving the expressions or equations are:

1. (x + a) (x + b) = x² + x(a + b) + ab
2. (a + b)² = a² + 2ab + b²
3. (a – b)² = a² – 2ab + b²
4. (a + b)² + (a – b)² = 2 (a² + b²)
5. (a + b)² – (a – b)² = 4ab
6. a² – b² = (a – b)(a + b)
7. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
8. (a + b)³ = a³ + b³ + 3ab(a + b)
9. (a – b)³ = a³ – b³ – 3ab(a – b) 
10. a³ – b³ = (a – b)(a² + ab + b²)
11. a³ + b³ = (a + b)(a² – ab + b²)
12. a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca))

Solved Examples on Algebraic Expressions

Problem 1: Find out the constant from the following algebraic expressions,

1. x³ + 4x² – 6
2. 9 + y⁵

Answer: 

Constants are the terms that do not have any variable attached to them, therefore, in the first case, -6 is the constant, and in the second case, 9 is the constant.

Problem 2: Find out the number of terms present in the following expressions,

1. 4x² + 7x – 8
2. 5y⁷ – 12

Answer:

Terms are separated by each other either by addition or subtraction sign. Therefore, in the first case, there are 3 terms and in the second case, there are 2 terms.

Problem 3: Simplify the algebraic term, z⁵ + z³ – y⁶ + 7z⁵ – 8y⁶ + 34 + 10z³

Solution:

In the expression, there are terms with the same power and same variable that are repeated, first bring them together,

(z⁵ + 7z⁵) + (z³ + 10z³) – (y⁶ – 8y⁶) + 34.

Now, simplify the expression,

8z⁵ + 11z³ – 9y⁶ + 34.

Problem 4: Add (13x² + 11), ( – 25x² + 26x + 42) and (–33x – 29).

Solution:

(13x² + 11) + ( – 25x² + 26x + 42) + (–33x – 29)

= 13x² – 25x² + 26x – 33x + 11 + 42 – 29

Now, add like terms

= –12x² – 7x + 24

Hence, (13x² + 11) + ( – 25x² + 26x + 42) + (–33x – 29) = –12x² – 7x + 24.

Problem 5: Solve (5x + 4y + 6z)² + (3y – 7x)².

Solution:

Given,

(5x + 4y + 6z)² + (3y – 7x)²

From algebraic formulae, we have

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

(a – b)² = a² – 2ab + b²

= (5x)² + (4y)² + (6z)² + 2(5x)(4y) + 2(4y)(6z) + 2(6z)(5x) + [(3y)² – 2(3y)(7x) + (7x)²]

= 25x² + 16y² + 36z² + 40xy + 48yz + 60zx +9y2 – 42xy + 49x²

Now, combine the like terms.

= 74x² + 25y² + 36z² – 2xy + 48yz + 60zx

Hence, (5x + 4y + 6z)² + (3y – 7x)² = 74x² + 25y² + 36z² – 2xy + 48yz + 60zx.

FAQs on Algebraic Expressions

Question 1: What are the types of algebraic expressions?

Answer:

The three main types of algebraic expressions are

• Monomial: A monomial is an expression that has only one non-zero term. 2xy, 5y3, 7a, 2b, etc are some examples of monomials.
• Binomial: A binomial is an expression that has two non-zero terms. For example, 2a + 3 has two monomials 2a and 3 and hence it is a binomial.
• Polynomial: A polynomial is an expression that has more than two non-zero, unlike terms. For example, x-4y+ 8z has three monomials x, 4y, and z, hence it is a polynomial.

Question 2: Are all algebraic expressions polynomials?

Answer:

Not all algebraic expressions are polynomials but all polynomials are algebraic expressions i.e. there exist some algebraic expressions that are not polynomials. Example: πx + 1 is an algebraic expression but not a polynomial.

Question 3: How are algebraic expressions defined?

Answer:

An algebraic expression is an idea of representing numbers using letters such as x, y, z, etc. without specifying their actual values. It is a mathematical statement that we get when arithmetic operations such as addition, subtraction, multiplication, or division are operated upon on variables and constants. In simple terms, an algebraic expression is a mathematical statement where variables have been combined using fundamental arithmetic operations.

Example: Ram’s age is three times the age of Akash, and their total age is 48. Express it as an algebraic equation.

Solution: Let the age of Akash be x. Then Ram’s age is 3x

3x + x = 48

4x = 48

Question 4: What are variables in an algebraic expression?

Answer:

In an algebraic expression, a variable is a symbol that doesn’t have a fixed value. For example, in the equation, 3x + 7 = 0, x is the variable that can take any value. In mathematics, a, b, m, n, x, y, z, etc are some examples of variables.