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A monomial is referred to as a polynomial, which is an algebraic expression with a single term but can have multiple variables and a higher degree too. For instance, 9x3yz is a single term, where 9 is the coefficient, 3 is the degree of a monomial, and x,y,z are the variables. Division of a monomial by another monomial is like any other operation performed in Mathematics. A monomial is an expression that consists of a single term and maybe in the form of numbers, variables, or variables with power. A number like 4 is a monomial in itself. Similarly, an alphabet x is a monomial. When we combine them like 4x, it happens to be a monomial. Let us learn more about monomials, their parts, and the division of one monomial by another monomial.
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What is a Monomial?
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Monomial has been defined as a term in Mathematics, which happens to consist of only one term. It is also called a power product as by nature it is a product of powers of all the variables having exponents which are not negative. In simpler words, it can be put that a monomial is an algebraic expression with a single term. It may contain more than one variable with different and higher degrees.
The word monomial was inspired and taken from the late Latin word “binomial”, which means 2. The prefix was changed from bi to mono, in order to justify the meaning. For example, 4x2yz is a monomial, where 4 is the coefficient, x, y, z being variables, and x has degree 2. A monomial normally performs functions similar to that of a polynomial, named addition, subtraction, multiplication, and division. A lot of arithmetic operations are done taking the help of monomials.
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Parts of a Monomial
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Like any other expression, a monomial also consists of parts that make it very important. These are:
- Variable - The variable present in a monomial denotes the letters present in the expression.
- Coefficient - Coefficient of a monomial stands for the number to which the variables have been multiplied.
- Degree - To find the degree of a monomial, you simply have to add up all the exponents of it.
- Literal part - The alphabets along with their respective exponent values are known as the literal part of a monomial.
Suppose, 49x2y is a monomial expression.
Here,
49 is the coefficient.
x and y play the role of variables.
2+1=3 is the degree.
x2y is the literal part.
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Dividing Monomials by Another Monomial
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Dividing a monomial by another monomial is an arithmetic operation commonly performed. The process is almost the same as the multiplication of monomials. This is done by writing the two different algebraic expressions in their expanded form and then omitting the ones which fall common to both.
At the time of dividing a monomial by another monomial, we follow the rule of dividing the variables and the coefficients. During the division of the exponents, they are subtracted by following the rules.
For example, we have m3 n2 mn
= m3n2mn
= m2n
Another example,
What will be the coefficient of polynomial 25x3yz3, after being divided with 5x3z3?
We can clearly see after dividing we will be left with 5yz. But as the question asks for just the coefficient, we will be left with 5 as the answer.
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Things to Remember
- Monomials are considered as algebraic expressions which consist of only one term. They are similar to polynomials and perform functions of addition, subtraction, division, and multiplication. 3xy can be called an example of a monomial expression.
- A monomial has four crucial parts. These are named as a coefficient, variable, degree, and literal part. The degree of a monomial can be obtained by just summing up the exponents present in that expression. For example, the degree of abcd is 1+1+1+1= 4 whereas the degree of a2bcd is 2+1+1+1=5.
- Dividing monomials with another monomial is as simple as any other mathematical operation. When we are dividing them, we must be concerned about exponents, negative exponents, and negative coefficients.
- During the division, we must separate and divide the coefficients first. After that, we should shift our focus to the variables. For example in the expression 10xy / 2x, we are left with 5y as the result. Division of monomials is easy and should not be confused with.
Sample Questions
Ques. What is a monomial? Give an example. (2 marks)
Ans. A monomial is basically a polynomial that consists of only one term. They can be numbers, variables, or both of them multiplied together. For example, 4ab is a monomial. Any number by itself can also be called a monomial like 27, 108, or 3.
Ques. Solve the given equation 42x2y. (3 marks)
Ans. At first, we will factorise the coefficient of the variables present,
We can write 42 as 2 3 7
After this, we can simplify x2y as,
x x y
Therefore, the factorization of the monomial 42x2y is,
2 3 7 x x y
Ques. What is the difference between a monomial, binomial, and trinomial? (2 marks)
Ans. A monomial is an algebraic expression consisting of a single term, whereas on the other hand, a binomial has two non-zero terms and a trinomial consists of a maximum of three non-zero terms. 2a2 can be called a monomial, 2a2+ b can be called a binomial, and lastly, 2a2+b+c can be called a trinomial.
Ques. What operations can be performed with monomials? (2 marks)
Ans. There are many arithmetic operations performed by a monomial. These may be addition, subtraction, division, or multiplication of two monomials respectively. For example, if we consider the addition of two monomials, 4xy + 18xy will give 22xy.
Ques. Factorize the given monomials. 1. 20z5 (2 marks)
Ans. At first, we will factorize the coefficient of the variable z, which is 20
We can write 20 as 2 x 2 x 5
After this, we can simplify z5 as,
Z x z x z x z x z
Therefore, the factorization of the monomial 20z5 is,
2 x 2 x 5 x z x z x z x z x z
Ques. Define the degree of a monomial. (2 marks)
Ans. The degree of a monomial refers to the sum of the exponents of the variables present in the monomial term. For example, in the expression 4xyz, the degree of expression is 1 + 1+ 1= 3. The degree of a non-constant is zero if the monomial has a constant value.
Ques. Give an example of a dividing monomial. (2 marks)
Ans. Let us take a monomial such as 100x2 and divide it by 10x. At first, we will divide the coefficients which are 100/10. This will give us 10. In our next step, we will divide the variables, that is x2/x. This will give us x. Our final answer will be 10x.
Ques. Is it possible to get a monomial upon adding two monomials? (2 marks)
Ans. During the addition of two individual monomials, if we consider adding the same literal parts, the result will be equivalent to a monomial. In case it is different, the result will be a binomial.
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