Monomial

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

Here are a few examples of monomials:

-3x2yz, a3, and pqr.

Parts

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 -3x2yz and identify its parts.

1. Constants are fixed values or numbers in a monomial. Here, the constant is -3.
2. 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.
3. Degree is the sum of exponents of all the variables in a monomial. The degree or order of -3x2yz is 2 + 1 + 1 = 4
4. Literals are the variables with their exponents. Here, the literals are x2, y, and z.

Find the degree of the following monomials
a) 8a4bc2
b) 5x2
c) 3y3
d) 12p

Solution:

a) The exponents are 4, 1, and 2, respectively.
Thus, the degree of 8a4bc2 is 4 + 1 + 2 = 7.
b) The exponent is 2.
Thus, the degree of 5x2 is 2, a quadratic monomial.
c) The exponent is 3.
Thus, the degree of 3y3 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) 5m2 + 2n
b) 8
c) p5
d) ${m^{-\dfrac{1}{2}}}$

Solution:

a) 5m2 + 2n contains two terms and is thus not a monomial.
b) 8 is a constant, a single term, and thus a monomial.
c) p5 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.

Factoring

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, 22u5v2w can be factored as 2 ✕ 11 ✕ u ✕ u ✕ u ✕ u ✕ u ✕ v ✕ v ✕ w.

Write the given monomial in factored form: 25xy3

Solution:

25xy3 = 5 ✕ 5 ✕ x ✕ y ✕ y ✕ y
Thus, 25xy3 can be factored as 5, 5, x, y, y, y.

Simplifying

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.

Subtracting

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

On subtracting 14p2q3r and 5p2q3r, we get

14p2q3r – 5p2q3r = (14 – 5)p2q3r = 9p2q3r, a monomial.

Multiplying

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 5a2 and 2a4, we get 10a2 + 4 = 10a6.

Dividing

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

On dividing 10p2q3r by 2pq2, we get ${\dfrac{10p^{2}q^{3}r}{2pq^{2}}}$

= ${\dfrac{10}{2}}$ ✕ p2 – 1 ✕ q3 – 2 ✕ r = 5pqr

Find the sum, difference, product, and quotient of the monomials 64x2y and 4yx2.

Solution:

On adding the given monomials, the sum is
64x2y + 4yx2 = (64 + 4)x2y = 68x2y
On subtracting, the difference is
64x2y – 4yx2 = (64 – 4)x2y = 60x2y
On multiplying, the product is
64x2y ✕ 4yx2 = (64 ✕ 4)x2+2y1+1 = 256x4y2
On dividing 64x2y by 4yx2, 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

Differences Between Monomial, Binomial, and Trinomial

Identify the monomials, binomials, and trinomials from the following.
a) a2b
b) x3 + y3
c) a3b2 + a2b2 + 11abc

Solution:

a) a2b contains one term, a monomial.
b) x3 + y3 contains two terms x3 and y3.
Thus, it is a binomial.
c) a3b2 + a2b2 + 11abc contains three terms a3b2, a2b2, and 11abc.
Thus, it is a trinomial.

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