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Algebra includes the operation of polynomials, equations and algebraic structures. Variables and constants are two notable aspects of algebra in general. Constant is a part of an algebraic equation which consists of only numbers while variables deal with letters. In case of constants, the value remains same while in the case of variable, the value can change from time to time.
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Key Takeaways: Algebraic Expressions, Algebraic Terms, Variables, Constants, Polynomials, Monomials, Binomials
What is an Algebraic Expression?
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There are two important notable things in algebra: algebraic expressions and algebraic terms. Algebraic term means the grouping of one or more factors, like 4n2, 2xy, -8x, etc. In this case, multiplication is the operation that connects the constants and variables. However, in case of algebraic expressions, a fixed value can be assigned to an algebraic term, and the result can be found out accordingly. For example, we can take an algebraic term arbitrarily, for example 6ab. It does not have a fixed value as of now. But if we assign values to x and y, say, for example, 2 and 3 respectively, we get 36. In this case, 6ab is equal to 36 and 2 and 3 are the factors.
Algebraic Expression
The factors can also be broken down into smaller factors. 6ab can take different values if we assign different values to a and b. Therefore, it can be stated that a term is a combination of constants and variables separated by multiplication or division. An algebraic expression is a combination of such terms basically.
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Types of Algebraic Expressions
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There are mainly three types of algebraic expressions: Monomial expression, Binomial expression and Polynomial expression.
Types of Algebraic Expressions
Monomial Expression
Expressions that contain only one algebraic term are referred to as monomial expressions. For example, 6x, 7ab, -14acd are all monomial expressions.
Binomial Expression
Algebraic expressions having two algebraic terms are known as binomial expressions. A few examples are 6 + 5x, 4a + b, etc.
Polynomial Expression
Algebraic expressions which have more than two algebraic terms are known as polynomial expressions. They also contain non-negative integral components of variables. An example of a polynomial expression is bx + 4y + 12d.
There are also two types of expressions:
- Numeric expressions which contain only fixed values like 10, -3, 55 and
- Variable expressions which contain variables and numbers like 4x+y, 3ac-22, etc.
What are Constants?
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In an algebraic expression, the terms which deal with only numbers and have fixed values are known as constants. This is the reason why they are called ‘constants’ since they remain the same and do not really change. There is no presence of any changed value which can alter the state of it being constant.
Constant Term
For example, we can take the number 30. The value is stated and no variable is associated with it, therefore it can confidently be said that the number is a constant. In the algebraic expression 3xy + 4, the value 4 is a constant, since it has no variable with it and it has its own definitive value. But the term 3xy has variables associated with it, which means that the value of this expression can change based on the values of x and y. Constants always have fixed numerical values.
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| Related Articles | ||
|---|---|---|
| Functions of Polynomials | Cubic Polynomials | |
| Substitution Method | Cross Multiplication | |
What are Variables?
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Unlike a constant, a variable consists of letters having no fixed value. In case of variables, the letters can represent any value and therefore, the algebraic expression can change accordingly.
For example, in the algebraic expression 4x + 3y = 0, the values x and y represent variables. X and y can take any values which will eventually satisfy the equation. More than one value can be substituted for the letters in an algebraic expression. Variables are represented by letters all the time.
Variables
Constants and variables are both used in algebra and the value of variables can be found out using constants. In an algebraic expression 4x+5, the term ‘5’ is a constant while the term ‘4x’ is a variable quantity. The value of ‘x’ is not known and can only be calculated through an equation. In this term, 4 is a coefficient of x.
Examples of Constants and Variables
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There are many examples we can consider to understand the differences between variables and constants. A few of them are stated below.
- In 6x, 6 is a constant while x is a variable.
- In -18pq, -18 is constant while p and q are variables.
- In 4a, 4 is a constant and ‘a’ is a variable. But 4 and ‘a’ together are a variable expression.
It can be noted that the product of constants and variables is always a variable.
Differences Between Constants and Variables
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The differences between constants and variables are stated below:
| Constants | Variables |
|---|---|
| In an algebraic equation, constants are numerical quantities which have fixed values. | In an algebraic equation, variables are numerical quantities which have values that change over time. |
| In case of constants, there can exist only one value. | In case of variables, there can be any value assigned to the representative of the value. |
| Constants are always written as numbers. | Variables are written in the form of letters. |
Things to Remember
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- Algebra includes the operation of polynomials, equations and algebraic structures.
- Algebraic term means the grouping of one or more factors to form a term, like 2xy, 6x etc.
- Equations, where there is only one algebraic term, are referred to as a monomial expression.
- Algebraic expressions which have more than two algebraic expressions are known as polynomial expressions.
- The terms which deal with only numbers and have fixed values are known as constants.
- In an arithmetic expression, a variable is a quantity which consists of letters having no fixed value.
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Sample Questions
Ques. Solve the expression 16x2 - 4x2 using the standard identity formula. (3 Marks)
Ans. The expression 16x2 - 4x2 can also be written as (4x)2 - (2x)2
Using the identity,
a2 - b2 = (a+b) (a-b)
(4x)2 -(2x)2 = (4x+2x) (4x-2x)
(4x)2 - (2x)2 = (6x) (2x)
(4x)2 - (2x)2 = 12 2
(4x)2 - (2x)2 = 24
Ques. Solve 4x + 6 = -14. (3 Marks)
Ans. 4x + 6 = -14
Putting 6 on the other end of the equation, we get
4x = -14 - 6
Or, 4x = -20
Dividing both sides by 4, we get
x = -5
Therefore, x = -5
Ques. Solve 5x = 4x + 7. (2 Marks)
Ans. 5x = 4x + 7
Putting 4x on the left side of the equation, we get,
5x - 4x = 7
Or, x = 7
Therefore, x = 7
Ques. Solve 7x + 5 = 6x + 2. (3 Marks)
Ans. 7x + 5 = 6x + 2
Taking 6x to the left side, we get,
7x + 5 - 6x = 2
Now, putting 5 on the right side, we get,
7x - 6x = 2 - 5
or, x = -3
Therefore, x = -3
Ques. Separate constants and variables from the following: 5, 6z, -12x, 7/5, 3/4xz, mn, 2p, 0, 3x/4, 7/12k, -pt/2m, ½ (2 Marks)
Ans. 5 is a constant.
6z is a variable.
-12x is a variable.
7/5 is a constant.
mn is a variable.
2p is a variable.
0 is a constant.
3x/4 is a variable.
-pt/2m is a variable.
½ is a constant.
Ques. Solve 8q - 5 = -4q + 7. (3 Marks)
Ans. 8q - 5 = -4q + 7
Taking -4q on the left side, we get,
8q - 5 + 4q = 7
Next, we put -5 on the right side
8q + 4q = 7 + 5
Solving both sides, we get,
12q = 12
Or, q = 1
Ques. State the coefficient of x and y in the algebraic expression 10x - 10y + 3. (2 Marks)
Ans. In the algebraic expression 10x - 10y + 3, 10 is the coefficient of x while -10 is the coefficient of y.
Ques. Find the numerical coefficients in the algebraic expression 12x2 + 5xy - 2x2y. (2 Marks)
Ans. In the algebraic expression 12x2 + 5xy - 2x2y, 12 is the coefficient of x2, 5 is the coefficient of xy and -2 is the coefficient of 2x2y.
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