Table of Contents

Last modified on January 11th, 2024

A polynomial containing a single non-zero term is called a monomial.

Here are a few examples of monomials:

-3x^{2}yz, a^{3}, and pqr.

It consists of either a fixed value, an unknown value, or a product of fixed and unknown values without other operators. Let us consider the monomial -3x^{2}yz and identify its parts.

**Constants**are fixed values or**Variables**are unknown values represented by letters or alphabets in a monomial. Here, the variables are x, y, and z. Also, the exponents of the variables must always be a whole number. Here, the exponents of x, y, and z are 2, 1, and 1, respectively, which are whole numbers.**Degree**is the sum of exponents of all the variables in a monomial. The degree or order of -3x^{2}yz is 2 + 1 + 1 = 4**Literals**are the variables with their exponents. Here, the literals are x^{2}, y, and z.

**Find the degree of the following monomials ****a) 8a ^{4}bc^{2}**

Solution:

a) The exponents are 4, 1, and 2, respectively.

Thus, the degree of 8a^{4}bc^{2} is 4 + 1 + 2 = 7.

b) The exponent is 2.

Thus, the degree of 5x^{2} is 2, a quadratic monomial.

c) The exponent is 3.

Thus, the degree of 3y^{3} is 3, a cubic monomial.

d) The exponent is 1.

Thus, the degree of 12p is 1, a linear monomial.

**Identify the monomials from the following expressions.****a) 5m ^{2} + 2n**

Solution:

a) 5m^{2} + 2n contains two terms and is thus not a monomial.

b) 8 is a constant, a single term, and thus a monomial.

c) p^{5} has one variable term p with a non-negative exponent 5, thus a monomial.

d) ${m^{-\dfrac{1}{2}}}$ contains a single term with a negative fractional exponent and is thus not a monomial.

To factor monomials, we individually factor all the coefficients and the literal parts. It is a way of expressing a monomial as a product of two or more monomials.

For example, 22u^{5}v^{2}w can be factored as 2 ✕ 11 ✕ u ✕ u ✕ u ✕ u ✕ u ✕ v ✕ v ✕ w.

**Write the given monomial in factored form: 25xy ^{3}**

Solution:

25xy^{3} = 5 ✕ 5 ✕ x ✕ y ✕ y ✕ y

Thus, 25xy^{3} can be factored as 5, 5, x, y, y, y.

The sum of any two monomials with the same literal parts is always a monomial.

Let us add two monomials, 4xyz and 10xyz.

Since the literal parts are xyz in both the given expressions, the sum is:

4xyz + 10xyz = (10 + 4)xyz = 14xyz.

The difference of any two monomials with the same literal parts is always a monomial.

On subtracting 14p^{2}q^{3}r and 5p^{2}q^{3}r, we get

14p^{2}q^{3}r – 5p^{2}q^{3}r = (14 – 5)p^{2}q^{3}r = 9p^{2}q^{3}r, a monomial.

Monomials with the same literal parts are multiplied by the laws of indices. However, monomials with different literal parts are multiplied by the law of integration. Thus, the product of two monomials is always a monomial.

On multiplying 4y and 4xz, we get 16xyz.

Again, on multiplying 5a^{2} and 2a^{4}, we get 10a^{2 + 4} = 10a^{6}.

While dividing the monomials with the same literal parts, we use the rules of indices.

On dividing 10p^{2}q^{3}r by 2pq^{2}, we get ${\dfrac{10p^{2}q^{3}r}{2pq^{2}}}$

= ${\dfrac{10}{2}}$ ✕ p^{2 – 1} ✕ q^{3 – 2} ✕ r = 5pqr

**Find the sum, difference, product, and quotient of the monomials 64x ^{2}y and 4yx^{2}.**

Solution:

On adding the given monomials, the sum is

64x^{2}y + 4yx^{2} = (64 + 4)x^{2}y = 68x^{2}y

On subtracting, the difference is

64x^{2}y – 4yx^{2} = (64 – 4)x^{2}y = 60x^{2}y

On multiplying, the product is

64x^{2}y ✕ 4yx^{2} = (64 ✕ 4)x^{2+2}y^{1+1} = 256x^{4}y^{2}

On dividing 64x^{2}y by 4yx^{2}, the quotient is

${\dfrac{64x^{2}y}{4yx^{2}}}$ = ${\dfrac{4\times 4\times 4\times x\times x\times y}{4\times y\times x\times x}}$ = 16

Basis | Monomial | Binomial | Trinomial |
---|---|---|---|

Definition | A polynomial containing only one term. | A polynomial containing two unlike terms. | A polynomial containing three unlike terms. |

Examples | 2x, 5p^{5}, 6pqr^{7}, and 8 | x + 7, 2p – 5q^{2}, 5m + 3t | x + 2y + 6z, a + b + 11 |

**Identify the monomials, binomials, and trinomials from the following.****a) a ^{2}b**

Solution:

a) a^{2}b contains one term, a monomial.

b) x^{3} + y^{3} contains two terms x^{3} and y^{3}.

Thus, it is a binomial.

c) a^{3}b^{2} + a^{2}b^{2} + 11abc contains three terms a^{3}b^{2}, a^{2}b^{2}, and 11abc.

Thus, it is a trinomial.