Basic Algebra: Monomials

Definitions

Monomials are algebraic expressions with a constant number multiplied by variables with non-negative integers exponents.

Example of monomials

2x^{3} y^{7} z^{12}    |   14x^{2} y^{3} z^{44}    |   -7x^{5} z^{11}

Example of non-monomials

2 frac{x}{y}    |    14 sqrt{x}    |    -7 x^5+2 z^{11}    |    3 x^{frac{3}{5}}

Monomials are written in their standard form the monomial looks compacted, and numbers are all grouped in a coefficient and each letter appears one time, with an eventual exponent. The numerical part is called Coefficient and the letter part is called “Literal part”. We can always turn a Non Standard form to its Standard form grouping the elements of the Coefficient and Literal part.

Standard form

20x^{3} y^{7} z^{12}

Non-standard form

20 x^{2}5xy^{5}2y^{2}z^{10}z^{2}

degree of a monomial

There are two distinguishable degrees for a monomial: the total degree and the degree of a specific variable (this might seem useless but in maths and physics, even if we have multiple variables sometimes we like to pick a specific one variable , which is defined by a letter, and consider its properties and not all the others).

Total degree

20x^{3} y^{7} z^{2}

The total degree is 3+7+2= 12

Specific Variable Degree

20x^{3}y^{7}z^{2}

The degree of x is 3, 7 for y, and 2 for z.

sum of monomials

The sum of a monomial is possible when we have two similar monomials.

Similar monomials

2x^{3} y^{7} z^{12}    |    5x^{3} y^{7} z^{2}

They have the same literal part

Non-similar monomials

20x^{3} y^{7} z^{12}    |    20x^{3}y^{3}z

They don’t have the same literal part

Similar monomials are these monomials with the same literal part, including their exponents.
The sum of these two monomials is the algebraic sum of the coefficients, while the literal part remains the same.

Sum of similar monomials

2x^{3}z^{2}+11x^{3}z^{2} = 13x^{3}z^{2}

We sum 2+11

sum of identical monomials

2x^{3}z^{2}+2x^{3}z^{2} = 4x^{3}z^{2}

We sum 2+2

Usually we’ll find different monomials in long expressions that we can group and sum.
$2x^{2}+3y^{2}z+5x^{2}+3x^{2}+2y^{2}z = (2x^{2}+5x^{2}+3x^{2})+ (3y^{2}z+2y^{2}z) = 10x^{2}+5y^{2}z$

Product of monomials

The product of two or more monomials doesn’t need similar monomials. It will change the resulting coefficient part and literal part: it is the multiplication of all the coefficients of the multiplied monomials and the multiplication of all the literal parts following the rules of exponents. This means the same letters with their exponents will appear with their exponents’ values summed. If a letter doesn’t appear twice, it will keep its exponent.

We try to multiplicate $3xy^{2}z^{3}·2x^{4}yz^{3}$

Coefficient part

3\cdot2=6

Coefficients are 2 and 3, we multiply them

Literal part

xy^{2}z^{3}·x^{4}yz^{3}=x^{(1+4)}y^{(2+1)}z^{(3+3)} = x^{5}y^{3}z^{6}

Letters are grouped, and their exponents are summed.

Examples of multiplications are:
$2xy·5xy = 10x^{2}y^{2} | 2x^{2}y^{3}z·5xy=10x^{3}y^{4}z | -frac{1}{4}xy·8z = -2xyz$

Power of a monomial

The power of a monomial is the power of the coefficient part and the power of the literal part.
As in the multiplication case, we still need to follow the rules of powers for the literal part: the power is the multiplication of the exponent of the letter for its power.

(2xy^{3})^{2}=2^{2}x^{1·2}y^{3·2} = 4x^{2}y^{6}

If we pay attention, we’ll figure out it happens for the coefficients too.
$(4x)^{2}=(2^{2}x)^{2} = 2^{2·2}x^{1·2}$

(x^{2}y)^{3}=x^{6}y^{3}

The coefficient part is represented by 1, so omitted. In fact, these all follow the rules of exponents.

Division of monomials

The monomial that is being divided is called the dividend, and the monomial that it is being divided by is called the divisor, like in the numbers’ case.

A Dividend can be divided by a divisor if and only if:

The Dividend and the Divisor have some letters in common
The letters which are not in common must be part of the Dividend
The exponents of the letters in common in the Dividend must be equal to or greater than the ones in the dividend

This happens because of the rules of the powers and monomials definition: a division between powers is the difference of the exponents. If an exponent must be a natural number in a monomial, the difference must return a non-negative integer number, and that’s why the exponents of the dividend must be greater or equal: the letters which are not in common must be part of the Dividend.

So, as in the multiplication case, we will divide the coefficients and the literal part separately.

We try to divide $3xy^{2}z^{3}$ by $2yz^{3}$

Coefficient part

3/2 = frac{3}{2}

Coefficients are 2 and 3, we divide them to get the quotient (or rational number) $frac{3}{2}$

Literal part

xy^{2}z^{3}/yz^{3}=x^{(1-0)}y^{(2-1)}z^{(3-3)}=x^{1}y^{1}z^{0}=xy

Letters are grouped, and their exponents subtracted: in the dividend we have exceeding letters and greater or equal exponents (when a letter is missing in the dividend, it means its exponent is zero).

What happens if we don’t follow the monomial division rules?

Let’s try it: we want to divide $6xy^{2}/2xy^{3}$

The result would be $frac{6}{2}x^{(1-1)}y^{(2-3)}=3x^{0}y^{-1}=3y^{-1}=frac{3}{y}$

If we want to divide $6y^{2}/2xy^{3}$ it would mean we want to divide $6x^{0}y^{2}/2xy^{3}$

The result would be $frac{6}{2}x^{(0-1)}y^{(2-3)}=3x^{-1}y^{-1}=frac{3}{xy}$

What does it mean?
The sum and division are not an internal operation of the set of monomials, while the multiplication and the power are internal operations.
When we have an operation and the result it’s still a monomial in every case, this is why we call this specific operation “internal”.
When we do sums and divisions, we don’t always get a monomial result.
That explains the rules about dividend exponents in division and the rule of the similar monomials in sums.

LCM

Lowest Common Multiple of Monomial

The LCM of a Monomial is similar to the LCM of constant numbers.
If we have two or more monomials, the LCM monomial is that resulting monomial with the property to have a coefficient part as LCM of the coefficients and the literal part as LCM of literal parts.

In particular, we take every number (or letter) once with the greatest exponent.

Coefficient part

The LCM of the absolute value of the numbers if they are integers
If there is at least one non-integer number (for example, $frac{2}{3}$ ), the LCM is always 1.

2xy, 3xy, 5xy. LCM is 2·3·5= 30
2xy, 20xy. The factors are $2, 2^{2}·5$ so the lCM is $2^{2}·5 = 20$
2xy, 3xy, $frac{2}{3}xy$ . LCM is 1, because we have a non-integer number.

Literal part

The LCM of the literal part is made of the single letters took once, but the ones with greater exponents.
The rules of exponentials are followed.

$x^{2}y, xy^{5}z.$ LCM is $x^{2}y^{5}z$ .
The value z is considered $z^{0}$

To get the LCM monomial, we mix these two elements together.

LCM of $2x^{2}y, 4xy^{5}z, 12z^{5} = 12x^{2}y^{5}z^{5}$

LCM of $2y, frac{1}{2}xz, 12z^{5} = xyz^{5}$ (1 is usually omitted).

GCF

Greatest Common Factor

The GCF is very similar to the LCM, but in this case we want a greatest common factor, so we’d pick the elements in common with the smaller exponent.

If we have two or more monomials, the GCF monomial is that resulting monomial with the property to have a coefficient part as GCF of the coefficients and the literal part as GCF of literal parts.

In particular, we take every number (or letter) once with the smallest exponent.

Coefficient part

The GCF of the absolute value of the numbers if they are integers
If there is at least one non-integer number (for example, $frac{2}{3}$ ), the GCF is always 1.

$2xy, 3xy, 5xy.$ GCF is 1
$2xy, 20xy.$ GCF is 2
$2xy, 3xy, frac{2}{3}xy$ . GCF is 1, because we have a non-integer number.

Literal part

We need to consider the exponents of the single letters in every monomial.
The rules of exponentials are followed.

$x^{2}y, xy^{5}z$ . GCF is xy.
$x^{2}yz^{3}, xy^{5}z$ . GCF is xyz.
$x,y,z.$ The GCF is 1 since the smallest has exponent zero. That’s why we said the letters which appear in every monomial.

To get the GCF monomial, we mix these two elements together.

GCF of $2x^{2}y, 4xy^{5}z, 12z^{5} = 2$ (2 is common and the smallest, for the literal part no variables appear in every monomial)

GCF of $2xy, frac{1}{2}xz, 12xz^{5} = x$ (1 is usually omitted).

When the literal part has no elements in common, we will declare 1 as GCF: why that?
We pick three monomials with no variables in common: x, y, z.

We can write $x=x^{1}y^{0}z^{0} | y=x^{0}y^{1}z^{0} | z=x^{0}y^{0}z^{1}$
GCF is automatically 1 when there is no letter that appears in every monomial since the GCF would be, following the logic rule: $x^{0}y^{0}z^{0} = 1·1·1=1$

Conclusions

This lecture is complete now, it will be followed by an Exercise lecture to practise what we have learned. If you have any questions about this topic please leave a comment on Youtube. This content is absolutely free, if you want to support me, you can subscribe to my youtube channel and follow me on Twitter. You are more than welcome to join my live streams on Twitch where I create my content and spend time with my community while gaming, producing art/writing or talking about science.