A monomial is a polynomial, which has only one term. As we know, the polynomials are the equations or algebraic expressions which consists of variables and coefficients and has one or more terms in it. Each term of the polynomial is a monomial. A polynomial involves the operation of addition, subtraction, multiplication, and non-negative integer exponents of variables.
A monomial is a type of polynomial, like, binomial and trinomial, which is an algebraic expression having only a single term, which is a non-zero. It consists of only a single term which makes it easy to do the operation of addition, subtraction and multiplication. It consists of either only one variable or one coefficient or product of a variable and a coefficient with exponents as whole numbers, which represent only one term, unlike binomial and trinomial, which consist of two and three terms respectively. It cannot have a variable in the denominator.
Like, 4x is a monomial example, as it denotes a single term. In the same way, 23, 4x2, 5xy, etc.are examples but 23+x, 4xy, 5xy-2 are not, as they don’t fulfil the conditions.
It is a product of powers of variables with non- negative integer exponents, such that, if there is a single variable x, then it has either a 1 or a power of xn of x, with n as positive integer and if for the product of multiple variables such that XYZ, then the monomials can be given in the form of xaybzc where a,b,c are non-negative integers.
Now, in terms of the coefficient, it is defined as the term with a non zero coefficient. The degree of a monomial is the sum of the exponents of all the included variables which forms monomials. For example, xyz2 have three degrees, 1,1 and 2. Therefore, the degree of xyz2 is 1+1+2 = 4.
Let us consider some of the variables and examples:
- p – One variable and degree is one.
- 5p2 – with 5 as coefficient and degree as two.
- p3q – with two variables and degree as 4(3+1).
- -6ty – two variable t and y and a coefficient -6
- Let us consider x3+3x2+4x+12 is a polynomial, where x3,3x2,4x and 12 are the single terms and called as monomials.
A binomial is a polynomial or algebraic expression, which has a maximum of two non-zero terms. It consists of only two variables.
Example are: 2x2 + y, 10p + 7q2, a + b, 2x2y2 + 9, are all binomials having two variables.
A trinomial is a polynomial or algebraic expression, which has a maximum of three non-zero terms. It consists of only three variables.
Example are: 2x2 + y + z, r + 10p + 7q2, a + b + c, 2x2y2 + 9 + z, are all trinomials having three variables.
Now hopefully, we have got the basic difference between Monomial, Binomial and Trinomial. Let us solve some problems based on monomial.
Question: Identity which of the following is a Monomial.
Answer: 3ab is a Monomial
Whereas 4b+c and 6x2+2y are binomials and a+b+c2 is a trinomial.
And all of these equations are called as a polynomial.