What is an Algebraic Expression?
An algebraic expression is a mathematical expression that consists of one or more variables, numbers in the form of coefficients and constants, and mathematical operations such as addition, multiplication, subtraction and division. An algebraic expression always has a finite number of terms. Let us learn about variables, coefficients, constants and terms in a little more detail.
What is a Variable?
A variable is a letter usually from the English alphabet that represents an unknown quantity. Different inputs associated with the variable will provide different outputs.
For example: x, y, z, and many other letters are commonly used as variables in algebraic expressions.
What are Coefficients?
The numerical factor of a term that contains a variable is known as its coefficient.
For example: In the expression 2x\(^2\) – 3x + 5, 2 is the coefficient of x\(^2\) and -3 is the coefficient of x.
What is a Constant?
A constant in any algebraic expression is a number whose value never changes.
For example: In the expression, x + 4, 4 is a constant.
What is a Term?
A constant and variable, or a variable with a coefficient in an algebraic expression, each of these parts of an expression is known as a term. An algebraic expression can have two types of terms, like and unlike terms.
Now let’s take a look at a few examples of algebraic expressions:
- x + 4 is an algebraic expression with two terms, x and 4, one variable, x, and a constant, 4.
- y\(^3\) + 2y\(^2\) + y – 9 is an algebraic expression with four terms, one variable, y and a constant, -9.
- z\(^2\) + y + x\(^3\) – 12 is an algebraic expression with four terms, two variables, x and y, and a constant, -12.
How do we write an Algebraic Expression?
There are some words and phrases which imply that they need to be represented as a math operation involving addition, subtraction, multiplication and division, as well as the inclusion of algebraic terms. These words or phrases can also be used to describe a relationship between variables and constants. Hence, these phrases can be transformed into an algebraic expression. This process of writing an expression from a phrase is known as writing an algebraic expression.
Let us take a look at a few examples on writing algebraic expressions:
1. 7 more than a number x – this is a phrase that needs to be represented as an expression.
‘More than’ implies an addition operation, so,
x + 7 is the required algebraic expression for the given phrase.
2. Similarly, for the phrase, 2 times b, the word times implies a multiplication operation or we need to find the product.
So the algebraic expression for this statement is, 2 × b.
Words or Phrases for some common Math operations
- Addition: more than, plus, sum or total of, increased by.
- Subtraction: fewer than, less than, take away from, difference, decreased by.
- Multiplication: times, product.
- Division: divided, quotient of.
Solved Examples
Example 1: Write the phrase as an algebraic expression:
7 less than a number x.
Solution:
x – 7 The phrase less than means subtraction.
Hence, x – 7 is the algebraic expression for the given phrase.
Example 2: Write the phrase as an algebraic expression:
The total of a number b and 21.
Solution:
b + 21 The phrase total of means addition.
Hence, b + 21 is the algebraic expression for the given phrase.
Example 3: Sam plants a tree that is 50 centimeters tall. The height of the tree increases by 40 centimeters each year. Write an expression that represents the height (in centimeters) after x years. What is the height after 12 years?
Solution:
Make a table showing the height of the tree each year for the initial few years. Use the results to write an expression and evaluate the expression when x=12. The height is increasing, so add 40 each year as shown in the table.
Years, x | Height (in centimeters) |
|---|---|
0 | 50 |
1 | 50 + 40(1) = 90 |
2 | 50 + 40(2) = 130 |
3 | 50 + 40(3) = 170 |
4 | 50 + 40(4) = 210 |
The required expression is 50 + 40x.
Evaluate 50 + 40x at x = 12.
50 + 40x = 50 + 40(12) = 530
So, the expression, 50+40x represents the height of the tree after x years and after 12 years the height of the tree is 530 centimeters.
An algebraic expression is a mathematical expression that consists of variables, coefficients and constants, along with mathematical operations such as addition, multiplication, subtraction, and division.
The terms in an algebraic expression with the same variables and with the same exponents are known as like terms.
The linear expression is an algebraic expression whose variables have the highest exponent of 1.
Non linear expressions are algebraic expressions in which the highest exponent of one or more variables is greater or lesser than 1.