research made by roberto orsaria and monica secco vocational shete high school for business,...
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Research made by Roberto Orsaria and Monica Secco
VOCATIONAL SHETE HIGH SCHOOL FOR BUSINESS, CATERING, HOTEL AND TOURISM SERVICES
“B. STRINGHER”
THE MONOMIALS
What are the monomials?The monomials are the smallest “bricks” with which the expression of literal calculation are built.
A literal expression is formed by a chain of monomials bound between them by the operation signs: +;-; ·; :.
+3a² 2ab - 5b3 + -6c
How can it be defined?
A monomial is a literal expression in which only multiplications and divisions between numbers and letters appear.
The following expressions are example of monomials:
¼x2y -¾a3bc2
-5xy2/z x -12a4
+3ab
The monomials are expressions formed by letters only:
a
x
y
or the expression formed by a single number:
+5
-3
¼
When can a monomial be defined as entire?
A monomial is defined entire when the denominator does not contain letters.As an example the following monomials are entire:
3a5b3
-2x3y¼ x
When can a monomial be defined as fractional?
A monomial is defined fractional when the denominator contains letters
As an example the following monomials are fractional:
2x/y3ab/c
1/x
In a monomial you can find:• A part made of numbers, which is said coefficient• A part made of letters
As an example in the monomial:
You can distinguish:the coefficent ¾ and the part made of letters a3b5
¾¾a3b5
a3b5
How is the degree of a monomial estimated?
The degree of a monomial is the sum of the exponents of all its letters.
3x2y3 degree: 2+3=5
23a2b4c degree: 2+4+1=7
-5xy degree: 1+1=2
Which is the degree of a monomial formed from a single number?
The degree of a monomial lacking the part made of letters is nought: in fact remembers that, any monomial with “a” (different from nought).
a0=0
The following monomials have nought degree:
-4 +5 +½
Which is the degree of a monomial with reference to one letter?
The degree of a monomial with refernce to a letter is the exponent of that letter.
As an example:3x3y5z
Degree with reference to y=5
Degree with references to x=3
Degree with references to z=1
When are two monomials equal?
Two monomials are equal if they have the same coefficient and the same part made by letters.
As an example the two monomials are equal:
+3xy2z +3zxy2
When are two monomials similar?
Two monomials are similar if they have the same part made by letters.
As an example the monomials are similar:
4a2b -7a2b +¼a2b
When are two monomials opposite?
Two monomials are opposite if they have the same part made by letters and opposite coefficients.
As an example the monomials are opposite :
+5xy -5xy
How can we work with the monomials?
Operations addition removal, multiplication, division and raising to power just of can be carried out with the monomials, like with the numbers, but it is in porheur.
How can be two monomials added?
As far as the sum of monomials you must keep in mind that:
two monomials can be added only if they are similar:
a monomial similar to the previous monomials in such case you will obtain and having as coefficient the algebrica sum of the coefficients.
An example:
The two monomials
+5a3b2 and -2a3b2
are similar and therefore they can be added and the monomial sum is :
(+5a3b2) + (-2a3b2 ) = (+5-2) a3b2 =+3a3b2
+5 a3b2 + -2 a3b2 = +3 a3b2
It is important to remember that:
Two monomials which are not similar cannot be added.
as an example the two monomials
+6xy e +3x2y
cannot be added.
How are multiply two monomials?
In order to multiply two monomials the literal coefficients and parts in letters must be multiplied between them, applying the property of the powers (that is adding the exponents).
+3 x2y · -2 x3y2 = -6 x5y3
How can you divide a monomial for an other?
In order to divide a monomial by another the numerical coefficients and the parts made of letters, must be divided between them applying the property of the powers (that is subtracting the exponents)
+12 a3b5 : +3 ab2 = +4 a2b3
How is the power of a monomial estimated?
In order to elevate to power a monomial the coefficient and every letter must be elevated to the given exponent applying in the monomial the property of the powers (multiplying the exponents)
+4 a3b52= +16 a6b10+42 a3·2b5·2
=
Examples:
(-2x2y3)3=(-2)3x2·3y3·3=-8x6y9
(-½bc4)2=(-½)2b2c4·2=+¼b2c8
(+3x-1y2)2= (+3)2x-1·2y2·2=+9x-2y4
