Algebraic Expressions are used to find answers for any Mathematical operation with variables, such as addition, subtraction, multiplication, or division. Algebraic expressions are classified into three types: monomial expressions, binomial expressions, and polynomial expressions. In addition, Boolean algebra is important in algebraic expressions. In the basics of algebraic expressions, an unknown value is represented by the letters x, y, and z. These letters are known as variables. An algebraic expression can contain variables as well as constants. They combine to create the algebraic expression.
What is an Algebraic Expression?
Algebraic expressions are created by combining variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. An algebraic expression is made up of terms; the equation may have one or more terms. Let's go through the basic terminology used in algebraic expressions.
Fixed numbers are referred to as constants in algebraic expressions. Variables are not associated with constants. For example, 3x - 1 has a constant of -1. Variables are the unidentified values in the algebraic expression. For example, y and z are variables in 4y + 5z. Coefficients are fixed values (actual numbers) that are assigned to variables. The variables are multiplied by them. For example, the coefficient of x2 in 5x2 + 3 is 5. A term can be either a constant or a variable, or a mix of the two. Addition or subtraction is used to separate each phrase. For example, the terms are 3x + 5, 3x, and 5.
Types of Algebraic Expressions
Algebraic expressions are classified according to the number of terms given in the below table:
|
Type of Algebraic Expression |
Definition |
Examples |
|
Monomial |
An algebraic expression with only one term is called a monomial algebraic expression. |
2xy, 5y3, 7a, 2b, etc |
|
Binomial |
An algebraic expression with two unlike terms or two monomials called a binomial algebraic expression. |
6x + 8y, 3a2 + 2ab, etc. |
|
Trinomial |
An algebraic expression with three unlike terms or three monomials is called a trinomial algebraic expression. |
x + y + z, 11x2 – 5y + z, etc |
|
Polynomial |
An algebraic expression that has two or more terms with non-negative integral exponents of a variable is known as a polynomial algebraic expression. |
3x + 4y + 5z, ax3 + bx2 + cx + d, etc |
|
Multinomial |
An algebraic expression that has one or more than one term. Here, the exponent of the variable can be negative also, is known as a multinomial algebraic expression. |
x + y-2, 5x2 + 6z-1 |
Simplifying Algebraic Expression
Simplifying algebraic expressions is simple and straightforward. To begin, lets understand what are like and unlike terms. Like terms have the same sign, whereas unlike terms have the opposite sign. To simplify an algebraic expression, first identify the terms that have the same power. Then, if the terms are like terms, combine them; if they are unlike, figure out the difference between the terms. The simplest version of an algebraic statement is one in which no power terms are repeated.
For example, 4x5 + 3x3 - 8x2 + 67 - 4x2 + 6x3, the same powers that are repeated are cubic and square, and when they are combined, the equation becomes 4x5 + (3x3 + 6x3) - (8x2 - 4x2) + 67. After simplifying the formula, the final result is 4x5 + 9x3 - 12x2 + 67. There are no repeated terms with the same power in this term.
Algebraic Expression - Addition
When two algebraic expressions are added, like terms are added with like terms directly, i.e., coefficients of the like terms are added. For example, let's add (25x + 34y + 14z) and (9x − 16y + 6z + 17):
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.
Algebraic Expression - Subtraction
To subtract one algebraic expression from another, we must add the second expression's additive inverse to the first expression. For example, let's subtract (5b2 + 6b + 8) from (3b2 − 5b):
(5b2 + 6b + 8) − (3b2 − 5b)
= (5b2 + 6b + 8) + (−3b2 + 5b)
= (5b2 − 3b2) + (6b + 5b) + 8 = 2b2 + 11b + 8
Algebraic Expression - Multiplication
When we execute a multiplication operation on two algebraic expressions, we must multiply each term of the first expression by each term of the second equation and then add all the results. For example, let's multiply (3x + 2y) with (4x + 6y − 8z):
(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
Algebraic Expression - Division
When dividing one algebraic expression by another, we can factorize both the numerator and the denominator, then cancel all the relevant terms and simplify the rest, or we can use the long division approach if we cannot factorize the algebraic expressions. For example, let's divide (x2 + 5x + 6)/(x + 2)
= (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 to solve expressions or problems are:
|
Formulas |
|
(x + a) (x + b) = x2 + x(a + b) + ab |
|
(a + b)2 = a2 + 2ab + b2 |
|
(a – b)2 = a2 – 2ab + b2 |
|
(a + b)2 + (a – b)2 = 2 (a2 + b2) |
|
(a + b)2 – (a – b)2 = 4ab |
|
a2 – b2 = (a – b)(a + b) |
|
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca |
|
(a + b)3 = a3 + b3 + 3ab(a + b) |
|
(a – b)3 = a3 – b3 – 3ab(a – b) |
|
a3 – b3 = (a – b)(a2 + ab + b2) |
|
a3 + b3 = (a + b)(a2 – ab + b2) |
|
3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca) |
Also Read: Algebraic Identities