A monomial in Maths is a type of polynomial that has only one single term. For example, 4p + 5p + 9p is a monomial because when we add the like terms it will obtain the result as 19p. Furthermore, 4x, 21x²y, 9xy, etc are monomials because each of these expressions includes only one single term. As we know, polynomials are the algebraic expressions or equations which include variables and coefficients and have one or more than one term.Each term of the polynomial is a monomial. A polynomial includes the operations of addition, subtraction, multiplication and non-negative integer exponents of variables. In this article, we will study polynomial, monomial definition, monomial problems monomial examples, degree of monomial etc.
Polynomials are expressions that include exponents that are added, subtracted, multiplied, or divided. There are different types of polynomial such as monomial, binomial, trinomial, and zero polynomial. A monomial is a type of polynomial with one single term. A binomial is a polynomial with a maximum of two unlike terms. A trinomial is a polynomial with a maximum of three unlike terms.
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A monomial is a special kind of polynomial, which is an algebraic-expression having only one single term which is non-zero. It includes only one single term which simplifies the operation of addition, subtraction or multiplication. It includes only a single variable or coefficient or product of variable or coefficient with exponents as a whole number, which denotes only one single term unlike binomial or trinomial which includes two or three terms. Monomials in Maths do not have a variable in the denominator.
For example, 3x is a monomial, as it represents only one single term. Similarly, 23, 3x², 7xy, etc.are examples of monomials but 13 + x, 3xy, 4xy -1 are not considered as monomials because they don’t satisfy conditions.
Monomial is a product of powers with non-negative integers. For example, if there is a one- variable p, then it will have a power either 1 or power of pⁿ with n as any positive integer and the product of the multiple variables such asPQR, then the monomial will be represented in the form of paqbrc. Here a, b and are non-negative integers. Monomial in terms of the coefficient is defined as the term with a non-zero coefficient.
A binomial is an algebraic expression or a polynomial which has a maximum two, unlike terms. For example, 2x + 5x² is a binomial as it has two unlike terms that is 2x and 5x².
A trinomial is an algebraic expression or a polynomial that has a maximum of, three unlike terms. For example, 2x + 5x² + 7x³ is a trinomial as it has three unlike terms that is 2x ,5x² and 7x³.
The degree of a monomial is the addition of the exponent of all the included variables which together forms a monomial. For example, pqr³ have 4 degrees 1,1,and 3. Therefore, the degree of pqr³ is 1 + 1 + 3 = 5.
x – Here, the variable is one i.e x and degree is also one.
6x² – Here, the coefficient is 6 and the degree is 2
x³y – Here, x and y are two variables and the degree is 4(3+1).
-6xy – Here, x and y are two variables and a coefficient is - 6.
Identify which one of the given below is a monomial.
4y + z
3x² + 3y
x + y + z²
Solution: 4xy is a monomial whereas 4y + z , 3x² + 3y are binomials and x + y + z² is a trinomial.
Solution: (x³x²) (yy³)
= x3 + 2 y1 + 3
Solve the monomial expression 16q + 7q - 2q-(-4q)
Solution: 23q - 2q + 4q
= 21q + 4q
(4x² + 3x -14 )+ (x³ - x² + 7x + 1)
Solution: (4x² + 3x -14 )+ (x³ - x² + 7x + 1)
Grouping the like terms, we get
x³ + (4x² - x²) + ( 3x+ 7x) + (-14 +1)
Simplifying the above expression we get,
= x³ + 3x² + 10x - 13
2. Which of the following has a similar value as ( t t t t t t) (zzzz)
1. Explain the Factoring of Monomials.
When two or more numbers are multiplied together, the result obtained is a single number. Factoring is a reverse process. In factoring, we initiate with a single number and express it as a product of two or more numbers. For example, products of 6 and 4 are 24. So, factoring 24, we get 6 * 4 = 24.
Every expression has two factors i,e 1 and the number itself. These are known as trivial factors. If monomial is a product of two or more numbers of variables, then it will have two factors one and itself.
For example: Let us calculate the factors of 49x⁴
Factors of 49 are 1, 7, and 49. The monomial factor of x³ is 1, x, x², x³ and x⁴.
Hence, the factors of monomial expression 49x³ are 12 products of a factor of 49 with a factor of x and the factors are 1,7,49, x, 7x, 49x, x², 7x², 49x², x³, 7x³, 49x³, x⁴, 7x⁴, 49x⁴.
2. What are the Steps to Find the Greatest Common Factors for the Monomials?
The greatest common factors of two or more monomials are defined as the product of two or more coefficients and the greatest common factors of the variables.
Steps to find the greatest common factor for the monomials.
Let us calculate the greatest common factor, for the monomial expression
5x³t and 10x²t².
The first step is to factor for each monomial
5x³t = 5*x*x*x*t , 100x²t² = 2 * 5* x *x * t
The next step is to select the factors between the monomials
5x³t = 5 * x * x * x * t , 100 x² t² = 2 * 5* x *x * t * t
The common factors are 5, x² and t.
Hence, the GCF between 5x³t and 10x²t². = 5x²t.
Note: The GCF of the monomials even includes the greatest common factors of the coefficients of the monomials. It also incorporates common variables raised to the least exponent of that variable found in the terms.