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A monomial is a polynomial that has just one term. A monomial is an algebraic expression with only one term, however, it can include several variables and a higher degree.
- Monomials are the building blocks of polynomials, and they are referred to as 'terms' when they appear in larger polynomials.
- In other words, every term in a polynomial is a monomial.
- For example, 2x3y is a single term with 2 as the coefficient, x and y as variables, and (3 + 1 = 4) as the degree of the monomial.
- Monomials, like polynomials, can perform many operations such as addition, subtraction, multiplication, and division.
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Key Terms: Polynomial, Variable, Coefficient, Literal part, Exponent, Monomial degree, Integer, Binomial, Trinomial, Factorization, Multiplication, Subtraction
What is a Monomial?
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A monomial is a polynomial that contains only a single non-zero term.
- Monomials consist of just a single term, making addition, subtraction, and multiplication easier.
- It is comprised of either a single variable, a single coefficient, or the product of a variable and a coefficient, with exponents that are whole numbers and represent just one term.
- Binomials and trinomials are also considered polynomials, with two and three terms, respectively.
- It cannot include a variable in the denominator.
Examples of MonomialBelow are some examples of monomials
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| Related Articles | ||
|---|---|---|
| Linear Equation | Addition and Subtraction of Integers | Binomial Theorem |
| Polynomials | Degree of polynomial | Factoring Trinomials Formula & Solved Examples |
How to Find a Monomial?
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A monomial can be easily identified using the following properties:
- A monomial expression must have one non-zero term.
- The variables' exponents must be nonnegative integers.
- There should be no variables in the denominator.
Let us examine the following cases to identify monomials.
| Expression | Monomial or not? | If not, then why? |
|---|---|---|
| 3x4y | Yes | - |
| 5x/2 | Yes | - |
| 2x3 + y | No | It has 2 terms i.e. 2x3 and y |
| 2x2/3 | No | The exponent of the variable is not an integer |
| 6y | No | The variable is an exponent |
| 7x/y | No | The denominator has a variable |
Parts of Monomial Expression
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The following are the parts of a monomial expression
- Variable: These are the letters that are present in a monomial expression.
- Coefficient: It is the number that is multiplied by the variable in the equation.
- Degree: It is referred to as the sum of the exponents present in the variables of the expression.
- Literal part: They are the alphabets that are present along with the exponent value in the expression.
Example of Parts of Monomial ExpressionExample: 3pq2 is a monomial expression. In the above example
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Degree of Monomial
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The monomial expression degree is formed by the addition of the variable's exponents in the given expression.
- When you calculate the monomial degree, exponent values of the variables are involved.
- Also, the exponent of 1 for the variables implied is included. This does not appear in the expression usually.
For example: 4xy2.
- Here, in the expression, the exponent value of 1 is not visible.
- Hence, the expression degree is 1 + 2 = 3.
- The order of the monomial is another name for the degree of the monomial expression.
Factorization of Monomial
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Monomial expressions can be factorized in the same way that numbers can.
- For example, the factorization of 12 is 3 × 4.
- The monomial expression can be written in the same way.
- Now, examine the monomial expression 16p3.
- First, factor the variable's coefficient, which is 16.
- The number 16 is factored as 2 × 2 × 2 × 2.
- Similarly, p3 factors as p × p × p.
- The factorization of monomial 16p3 is 2 × 2 × 2 × p × p × p.
Operations on Monomial
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Some operations performed on the monomial expression are mentioned below:
- Subtraction of two monomials
- Addition of two monomials
- Division of two monomials
- Multiplication of two monomials
Subtraction of two monomials
The subtraction of two monomials with the same literal part produces a monomial expression.
For example: 9pqr – 3pqr = 6pqr
Addition of two monomials
The subtraction of two monomials with the same literal part produces a monomial expression.
For example: 7xy + 6xy = 13xy
Multiplication of two monomials
Multiplying two monomials will also produce a monomial.
For example: x2 and x3 = (x2)( x3) = x2 + x3 = x5.
Division of two monomials
When dividing two monomials with the same variables, subtract the exponent value of each variable.
For example: p9 by q3 = (p9) / (q3) = p9 – 3 = p6.
Difference Between Monomial, Binomial, and Trinomial
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The following are the differences between Monomial, Binomial, and Trinomial
| Monomial | Binomial | Trinomial |
|---|---|---|
| A monomial is an expression having only a single term. | A binomial is a polynomial or algebraic expression that has a maximum of two non-zero terms. | A trinomial is a polynomial or algebraic expression that contains a maximum of three non-zero terms. |
| For example: 3x, 5y, 3z, 3x2, 9xyz, etc., are monomials | For example: 5x3 + y, 19p + 73q2, x + y, 2x3y2 + 7, are all binomials | For example: 3x2 + y + z, r + 7p + 7q3, a + b + c, 2x2y3 + 3 + z, are all trinomials |
Things to Remember
- A monomial is a polynomial that contains only a single non-zero term.
- Binomials and trinomials are also considered polynomials, with two and three terms, respectively.
- The degree is referred to as the sum of the exponents present in the variables of the expression.
- The coefficient is the number that is multiplied by the variable in the equation.
- The subtraction of two monomials with the same literal part produces a monomial expression.
- Multiplying two monomials will also produce a monomial.
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Sample Questions
Ques. Factorize the given monomial 12y2. (2 Marks)
Ans. Given that,
Monomial:12y2.
Factorization of the coefficient of the variable, y. (i.e.) 12.
Therefore, 12 can be factorized as 2×6.
y2 is factorized as y × y.
Therefore, the factorization of the monomial 12y2 is 2×6×y×y.
Ques. Is it possible to acquire a monomial term by adding two monomials? (2 Marks)
Ans. The sum results in a monomial when two monomials of the same literal parts are added. However, the result is binomial when two monomials of different literal parts are added.
Ques. Give solution for the monomial expression 15q + 8q - 2q- (- 4q) (2 Marks)
Ans. Given that,
= 15q + 8q - 2q-(-4q)
= 21q + 4q
= 26q
The answer is 23q - 2q + 4q
Ques. What is the degree of a monomial? (2 Marks)
Ans. The degree of a monomial can be explained as the sum of the exponents of the variables that are contained in the monomial.
Ques. How can you identify a monomial expression? (2 Marks)
Ans. The monomial expression has an addition or subtraction operator. It is usually a constant term. Otherwise, it has variables with coefficients as well as exponents.
Ques. Solve the given equation (4x2 + 3x -14 )+ (x3 - x2 + 7x + 1) (2 Marks)
Ans. (4x2 + 3x -14 )+ (x3 - x2 + 7x + 1)
By grouping,
x3 + (4x2 - x2) + ( 3x+ 7x) + (-14 +1)
Therefore, by simplification,
= x3 + 3x2 + 10x – 13
Ques. (x3y) (x2y3) (2 Marks)
Ans. Given that,
(x3x2) (yy3)
= x3 + 2 y1 + 3
= x5y4
Therefore the answer is x5y4
Ques. Factorize the given monomial expression: 10ab. (2 Marks)
Ans. In 10ab, the coefficient’s prime determinants are 10 are 2 and 5.
Thus, the variable part of the expression 'ab' is simplified as a × b.
Hence, the factorization of the given monomial is 10ab = 2 × 5 × a × b.
Ques. How do you simplify (2p2)(4x+p)? (2 Marks)
Ans. 2p2 is distributed to each term in the parentheses in the other given polynomial.
= 2p2(4x) = 8xp2
= 2p2(p) = 2p3
Therefore, by simplifying
= The answer is (2p2)(4x+p) = 8xp2 + 2p3
Ques. Factorize the given monomial 9b3. (2 Marks)
Ans. We know that,
9b3 is a monomial.
Therefore, 9 can be factorized as 3 × 3.
B3 is factorized as b × b × b.
Therefore, the factorization of the monomial 9b3 is 3 × 3 × b × b × b.
Ques. Simplify the given expression (7x)(7t − 3x3 + 2) (3 Marks)
Ans. We know that,
(7x)(7t−3x3+2)
When 7y is distributed to each term in the parentheses in the given other polynomial
= 7x (7t)= 49xt
On simplification,
= 7x (−3x3)= −21x4
= 7x (2)= 14x
By simplifying the result,
= (7x)(7t−3x3+2)= 49xt−21x4+14x
Therefore the answer is (7x)(7t−3x3+2)= 49xt−21x4+14x
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