The basic concept of algebra taught us how to express an unknown value using letters such as x, y, z, etc. These letters are termed here as variables. this expression can be a combination of both variables and constants. Any value that is placed before and multiplied by a variable is termed a coefficient. An idea of expressing numbers using letters or alphabets without specifying their actual values is defined as an algebraic expression.
Algebraic Expression
An expression that is made up of variables and constants along with algebraic operations such as addition, subtraction, etc. is termed an algebraic expression. These Expressions are made up of terms. Algebraic expressions are the equations when the operations such as addition, subtraction, multiplication, division, etc. are operated upon any variable. A combination of terms by the operations such as addition, subtraction, multiplication, division, etc is termed as an algebraic expression (or) a variable expression. Examples: 2x + 4y – 7, 3x – 10, etc.
The above expressions are represented with the help of unknown variables, constants, and coefficients. The combination of these three terms is termed as an expression. Unlike the algebraic equation, It has no sides or ‘equals to’ sign.
Types of Algebraic expression
There are three types of algebraic expressions based on the number of terms present in them. They are monomial algebraic expressions, binomial algebraic expressions, and polynomial algebraic expressions.
- Monomial Expression: An expression that has only one term is termed a Monomial expression. Examples of monomial expressions include 5x4, 2xy, 2x, 8y, etc.
- Binomial Expression: An algebraic expression which is having two terms and unlike are termed as a binomial expression. Examples of binomial include 5xy + 8, xyz + x2, etc.
- Polynomial Expression: An expression that has more than one term with non-negative integral exponents of a variable is termed a polynomial expression. Examples of polynomial expression include ax + by + ca, 3x3 + 5x + 3, etc.
Other Types of Expression
Apart from monomial, binomial, and polynomial types of expressions, there are other types of expressions as well that are numeric expressions, variable expressions.
- Numeric Expression: An expression that consists of only numbers and operations, but never includes any variable is termed a numeric expression. Some of the examples of numeric expressions are 11 + 5, 14 ÷ 2, etc.
- Variable Expression: An expression that contains variables along with numbers and operations to define an expression is termed A variable expression. Some examples of a variable expression include 5x + y, 4ab + 33, etc.
Some algebraic formulae
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
(a + b)(a – b) = a2 – b2
(x + a)(x + b) = x2 + x(a + b) + ab
(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)
Evaluate 2st2 – 52 for s = 4 and t = 8
Solution:
Given expression: s = 4 and t = 8
= 2st2 – 52
Now put value of s and t
= 2 (4)(8)2 – 52
= 8(64) – 25
= 512 – 25
= 487
Some Questions
Question 1: Solve for x: 5x – 50 = x + 3x
Solution:
5x – 50 = x + 3x
5x – 50 = 4x
5x – 4x = 50
x = 50
Question 2: Simplify (4x – 5) – (5x + 1)
Solution:
Given that, (4x – 5) – (8x + 1)
Step 1: Remove parentheses and apply the signs carefully.
= 4x – 5 – 8x – 1
Step 2: Bring like terms together
= 4x – 8x – 5 – 1
Step 3: Now add or subtract the like terms
= -4x – 6
= -2(2x + 3)
So the final result is -2(2x + 3)
Question 3: Solve for the value of t: 31 + t = 4 (t – 3) + 22.
Solution:
Given: 31 + t = 4 (t – 3) +22
31 + t = 4 (t – 3) + 22
31 + t = 4t – 12 + 22
31 + t = 4t + 10
31 – 10 = 4t – t
21 = 3t
t = 21/3
t = 7
So, the value of t is 7
Question 4: Solve for y in the equation: -1/y = -0.25x2 – 7.5
Solution:
Given: -1/y = -0.25x2 – 7.5
Multiply both sides by y
⇒ (-1/y)(y) = y(-0.25x2 – 7.5)
⇒ -1 = -0.25x2y – 7.5y
⇒ -1 = y(-0.25x2 – 7.5)
⇒ -1 = -y(-0.25x2 – 7.5)
⇒ y = 1/ (-0.25x2 – 7.5)