Factoring Polynomials using Greatest Common Monomial Factor
Summary
TLDRThis educational video script introduces grade 8 students to the concept of factoring polynomials using the greatest common monomial factor (GCMF). It explains the process of identifying factors and GCMF through examples with numbers and algebraic expressions. The script also covers prime factorization and demonstrates how to apply these techniques to factor binomials and polynomials, including the difference of cubes and sum of cubes, with the aim of simplifying complex expressions for easier understanding.
Takeaways
- đ The lesson is focused on factoring polynomials using the greatest common minimal factor, also known as the greatest common factor (GCF) or greatest common monomial factor (GCMF).
- đą A factor is defined as a number or algebraic expression that divides another number or expression evenly with no remainder.
- đ° Examples are given to illustrate how to find the factors of numbers such as 20 and 10, and then identify their greatest common factor (GCF).
- đ The script introduces the concept of prime factorization as a method to find the GCF of algebraic expressions, which involves breaking down numbers into their prime factors.
- đ The greatest common monomial factor (GCMF) is explained as the common factor with the smallest exponent for variables in algebraic expressions.
- âïž The process of dividing each term of a polynomial by the GCMF to find the other factors is described, using examples such as 6x + 3x^2 and 12x^3y^5 - 20x^5y^2z.
- đ The importance of identifying common factors in polynomials is emphasized to simplify the expression and make factoring easier.
- đ The script provides a step-by-step approach to factoring polynomials, including dividing each term by the GCMF and simplifying the result.
- đ The concept of exponents is applied when finding GCMF, where the variable with the smallest exponent is chosen for the factor.
- đ The script also covers how to handle terms without common variables or coefficients in the process of finding the GCMF.
- đ The lesson concludes with a reminder to check the factored form by multiplying the factors to see if they yield the original polynomial, and a teaser for upcoming lessons on additional factoring techniques.
Q & A
What is the definition of a factor in mathematics?
-A factor is a number or algebraic expression that divides another number or expression evenly with no remainder.
What are the factors of 20?
-The factors of 20 are 1, 2, 4, 5, 10, and 20.
How do you find the greatest common factor (GCF) of two numbers?
-You list all the factors of each number and find the greatest factor that is common to both numbers.
What is the GCF of 20 and 10?
-The GCF of 20 and 10 is 10.
What is the greatest common monomial factor (GCMF) and when is it used?
-The GCMF is the greatest common factor of algebraic expressions or monomials, and it is used when dealing with polynomials or expressions that contain variables.
How does the method of prime factorization help in finding the GCMF of algebraic expressions?
-Prime factorization breaks down the numbers into their prime factors, making it easier to identify and align the common factors in the expressions.
What is the GCMF of 4x^3 and 8x^2?
-The GCMF of 4x^3 and 8x^2 is 4x^2.
How do you factor a polynomial using the GCMF method?
-You find the GCMF of the terms in the polynomial, divide each term by the GCMF, and then multiply the GCMF by the resulting factors.
What is the factored form of 6x + 3x^2 using the GCMF method?
-The factored form of 6x + 3x^2 is 3x(2 + x).
What is the GCMF of 12x^3y^5 - 20x^5y^2z and how is it used to factor the expression?
-The GCMF of 12x^3y^5 - 20x^5y^2z is 4x^3y^2. It is used to divide each term and find the remaining factors, resulting in the factored form 4x^3y^2(3y^3 - 5x^2z).
How does the script differentiate between the GCF and GCMF?
-The script differentiates by using the term GCF for numbers and GCMF for algebraic expressions or monomials, emphasizing the variable with the smallest exponent in the GCMF.
Outlines
đ Introduction to Factoring Polynomials
This paragraph introduces the concept of factoring polynomials for 8th-grade mathematics. The focus is on using the greatest common minimal factor to factor polynomials. It starts by defining a factor as a number or algebraic expression that divides another evenly with no remainder. Examples given include finding the factors of 20 and 10, and then determining the greatest common factor (GCF) of these numbers. The GCF is identified as the largest number that divides both numbers without a remainder. The paragraph also introduces the concept of the greatest common monomial factor (GCMF) for algebraic expressions, which is similar to GCF but applies to variables and monomials.
đ Prime Factorization and GCMF
The second paragraph delves into the method of prime factorization to find the greatest common monomial factor (GCMF) of algebraic expressions. It explains that prime numbers, which have only two factors (one and itself), are used in this method. The process involves breaking down expressions into their prime factors and then identifying the common factors between two or more expressions. Examples are given to demonstrate how to factor expressions such as 4x^3 and 8x^2, and 15y^6 and 9z, by finding their GCMF and then dividing each term by this factor to simplify the expression.
đ Factoring Polynomials Using GCMF
This paragraph continues the discussion on factoring polynomials by applying the concept of the greatest common monomial factor (GCMF). It provides a step-by-step approach to factor binomial expressions like 6x + 3x^2 by identifying the GCMF from both the numerical coefficients and the variables, focusing on the variable with the smallest exponent. The method involves dividing each term of the polynomial by the GCMF and then simplifying to find the remaining factors. An example is given to illustrate the process, which results in the factored form of the polynomial.
đ Advanced Factoring with GCMF
The fourth paragraph presents a more complex example of factoring polynomials using the greatest common monomial factor (GCMF). It involves a trinomial expression, 12x^3y^5 - 20x^5y^2z, and demonstrates how to find the GCMF by examining both the numerical coefficients and the variables. The process includes dividing each term by the GCMF and simplifying to find the remaining factors. The example concludes with the factored form of the polynomial, showcasing the application of the GCMF in a more advanced context.
đ Factoring Trinomials and Beyond
This paragraph extends the factoring technique to trinomial expressions, specifically 28x^3z^2 - 14x^2y^3 + 36yz^4. It discusses the process of identifying the GCMF among the terms, which involves examining both the numerical coefficients and the variables. The paragraph emphasizes the importance of recognizing which variables are common to all terms and which are not. After determining the GCMF, the method involves dividing each term by this factor and simplifying to find the remaining factors, leading to the factored form of the trinomial.
đ Factoring with Non-Perfect Cubes
The final paragraph addresses the factoring of expressions that do not represent perfect cubes, such as 5h + 40hk^3. It clarifies that these expressions cannot be factored as sums of cubes and instead requires finding the greatest common monomial factor (GCMF). The process involves dividing each term by the GCMF, which in this case is 5h, and then simplifying to find the remaining factors. The paragraph concludes with an example that demonstrates this factoring method and hints at further lessons on factoring techniques.
Mindmap
Keywords
đĄFactoring Polynomials
đĄGreatest Common Factor (GCF)
đĄMonomial
đĄPrime Factorization
đĄVariable
đĄExponent
đĄBinomial
đĄTricomial
đĄSum and Difference of Cubes
đĄPerfect Cube
Highlights
Introduction to the concept of factors in mathematics, defining a factor as a number or algebraic expression that divides another evenly with no remainder.
Listing method for finding factors of numbers, demonstrated with examples of numbers 20 and 10.
Explanation of the greatest common factor (GCF) and its significance in mathematics.
Identification of the GCF for 20 and 10 as 10, showcasing the process of elimination to find the greatest common factor.
Introduction of the term 'Greatest Common Monomial Factor' (GCMF) for algebraic expressions.
Transition from GCF to GCMF when dealing with algebraic expressions, using the example of 4x^3 and 8x^2.
Prime factorization method for finding GCF, explained with the example of 4x^3 and 8x^2.
Finding the GCMF of polynomials using prime factorization, demonstrated with 15y^6 and 9z.
Method for factoring polynomials using GCMF, illustrated with the binomial 6x + 3x^2.
Tip on selecting the variable with the smallest exponent when finding GCMF for variables.
Example of factoring a polynomial with multiple terms, 12x^3y^5 - 20x^5y^2z.
Process of dividing each term by the GCMF to find the other factors in a polynomial.
Explanation of how to handle terms without a common variable or exponent when factoring.
Factoring a trinomial polynomial, 28x^3z^2 - 14x^2y^3 + 36yz^4, using GCMF.
Illustration of the prime factorization for numbers in a polynomial to find the GCMF.
Final example of factoring a binomial with terms 5h and 40hk^3, identifying the GCMF as 5h.
Factoring the remaining terms after identifying the GCMF, using the example of 1 + 8k^3.
Conclusion of the lesson with a teaser for upcoming lessons on additional factoring techniques.
Transcripts
good day everyone
and today is the first day when you're
going to
answer your module so
the first topic for grade 8 mathematics
is about factoring polynomials so this
time
i'm going to discuss to you how to
factor polynomials
using greatest common minimal factor
but let us recall first what
is factor
what do you mean by factor when you say
factor
that is a number or algebra expression
that divides another number or
expressions
evenly that is with no remainder
okay say for example 20
so we're going to search for numbers
for a number or numbers that divides the
another number
or expressions or these 20 evenly with
no remainder
okay so what are those so
let's list all the factors of
20
okay of course one
why one because since we when you're
going to divide 20 by 1
that is 20 so there is no remainder
how about 2 okay 2. so when you divide
20 by 2 that is 10 so
still 2 is a factor of twenty
okay next four why
since twenty divided by four that is
five that is
also a whole number no remainder
next five okay when you
when you divide 20 by 5 that is
4 and x 10
so 20 divided by 10 is 2. so there is no
remainder again
and last one is 20. so the twenty
divided by twenty
that is one so the factors of
twenty are one two
four five ten and twenty
let's have another example how about
10 let's find
the factors of 10
okay so again factors is a number or a
expression that divides another number
or expressions even liters
with no remainder of course
one so okay um one
is a factor to any expression or to any
number actually class
okay two yes two ten divided by two that
is five
five ten divided by 5 is 2
there's no remainder and lastly 10.
the factors of 10 are 1 2 5
10. so this method is listing method
release
all the factors okay
from these two numbers 20 and 10 we're
going to look at the
common factor that is the greatest
common factor
i guess everybody is very
they know or you know what is greatest
common factor
greatest common factor so look for
the number the co the common factor is
the greatest
so here so one is
the common factor for 20 and 10 2
five and one two five
and ten that is their common
they are the common factors but we're
going to search
for the greatest so obviously
the gcf this common factor is everywhere
it's ever basic evasion is gcf
the gcf of 20 and 10
is 10 okay
that's it 20 20 and fourteen
and ten is ten
okay again the gcf of twenty and 10
is 10. okay how about
if we're going to get
the gcf of 4x cubed
and eight x squared
so as you notice there uh there is x
cubed and x squared so we're not dealing
here purely numbers
so we're dealing here algebraic
expressions
or variables so since we're dealing with
that algebra expressions
so we're not going to use the word gcf
instead we're going to use g c
and f k g
g that is greatest c
for common m for monomial
f that is factor so m
is just the new word here so dcmf is the
greatest
common monomial factor okay
a while ago we use listing method in
finding the gcf of 20 and 10.
here we can do that also but
we're where it's really time consuming
we're running out of time or we're going
to consume
um more time so i'm going to introduce
to you
a new or
this is not you this was introduced to
you when you were elementary
new this method this is
prime prime factorization
okay from the word prime so
we're going to look factors that are
prime
or prime numbers so what is this prime
number
so to recall prime numbers are numbers
whose factors are only one and itself
example two two factors of two are only
two and one so two on itself two and
no one and itself so one and two three
three the factors of three are only
three and one
okay let's do that
four x cubed
okay four look for the prime factors
that would be
two times two x cubed
that would be x times x
times x so x is multiplied three times
so that would be that would result to x
cube
next eight x squared
so eight that is three two so two
times two okay there is times two
but in prime factorization the tip class
you need to align those factors that are
similar
so since here it's x here so it's not
this that's not the same with two or
different from two
so you write another two here so times
okay how about x squared
x squared is just times x so you're
right here
times x okay
and then you look for their
common factors so here too
okay and now i crush it so that
it's simple it's it's it's
more simple for us to determine
or to crash out the used common factor
okay another
common another two times
another common x another common
x but here it is x
is no common with x eight x squared
this also two has no common with 4 x
cube
and then so you multiply
common factors common prime factors
2 times 2 that is 4
x times x is x squared
okay so meaning
the g c m f of 4 x
cubed and 8 x squared is
4 x squared another example in finding
the gcmf or the greatest common factor
k let's have here 15
y to the power of 6 and
i'm sorry we're not going to write 9
and 9 z okay
sub it so we're going to use the prime
factorization method since this thing
method is really hard
so severe 15
15 y the power six
so 15 that is so find the prime factors
is 3 times y both 3 and 5 are prime
numbers
3 times 5 and y divided by 6 is
y multiplied six times so
y times y times
y times y times
y times y okay
that's it how about for nine z
okay nine z
okay four nine oh that's obviously
the prime factors is r
nine three multiplied by another three
so that's three since
after three four fifteen white barbie
six is five
so you will put your three and then for
z
there is no zero since they're y's
so multiply by z now look for
look for their common factor
so it's very clear
the common factor is only three so
meaning
the gcm f of 15 y to the power
6 and 9 z is
three okay we're done finding the gcf or
the gcm f
of two terms or two monomials now let's
have
polynomials so you're going to find the
gcmf
so let's have your number one
six x plus
three x squared
so as you notice there are two terms
here this this is binomial
so what we're going to do is you look
for their
of course gcmf so you have
let's start with the number
six and three
they're common or they're gcf or i mean
yeah you see f
is three okay
for x and x squared their gcm f
is also x
tips or um yeah i'll give you a tip
so when you're going when you're given
with variables
the gcf there is the variable with the
smallest exponent
for so for x and x squared that's x
okay and then you look for
the other factor so 3x is obvious
obviously the gcmf or the greatest
common monomial factor
but we're going to look also for the
other factor
how so you just divide
each term so yeah there is 6x
you divide it by the gcmf which is the
3x plus 3
x squared divided by
the gcm x also this 3x
okay next
copy 3x
so now you have where 6x divided by 3x
so 6 divided by three that is two
x divided by x that is
one okay
way one okay for example five divided by
five
you're not going to say zero five
divided by five
is one so if you have x divided by x
that is one
so two times one that's obviously
that's obviously 2. next
3 x squared divided by 3 x
3 divided by 3 that is 1 also
x squared divided by x
squared i am in x so that is x
y what i did here is
just subtract the exponent k
recall right here i recall
and i know rules and i mean loss and
of exponent so if you have your a
8 the power of m over or divided by
a to the power of n so what
are you going to do here is just you
just subtract the exponent so
m minus n okay
recall so here x squared
so 2 for the x class there is no
exponent
that you see for x but obviously the
exponent
for x is one understood nanga
one an exponent so two minus one that is
one or it's just
x why is it i did not put one
for the numerical coefficient of x
of course it's obvious the numerical
coefficient of x
is one no need to write it
okay so again
the factors of six
x plus three
x squared are three x
plus i mean multiplied
multiplied by the quantity of two
plus x okay
class you can check actually your
answers
okay i'll give you a tip here how
so three x you just multiply
the given factors or the factors that
you
that you have that you got
okay you multiply so here you distribute
3x times 2 that is of course
6x 3x times
x that is 3x squared
so what did you know not notice
so you have you come up with the product
of the given polynomial
let's have another example in factoring
factoring polynomials using
gcmf okay so you have your number two
12x cubed y to the power of 5
minus 20 x to the power of 5
y squared z okay
so you copy 12 x cubed
y to the power of 5 minus 20
x to the power of 5 y squared
z okay so you look for the gcmf so let's
start with the numbers
so for 12
so for 12 and 20
obviously their gcmf is
yes four
okay four since when you divide twelve
by four that's three twenty divided four
is five so
that's a whole number it's no remainder
okay for
x cubed and x to the power of five
okay x cubed and x five so
i said it a while ago that the tip
for finding the gcf or gcmf
is to select the variable was the
smallest
exponent so that is
that is x cube
okay x cube now for y
to power 5 and y squared so that's
obviously
y squared okay how about z
as you notice a 12
x cubed y by 5 there is no z so meaning
that's obvious that z
is the not common factor for the two
terms
okay so meaning 4 x cubed y
squared is the gcm f
now let's look for the other factor
how so just divide each term by the gcm
f
4x cubed y squared so you have 12
x cubed y
to the power of 5 divided
by 4 x cubed
y squared minus
20 x to the power of 5
y squared z
divided also with the gcm f 4
x cubed y squared
okay okay let's do it
four x cubed y
squared okay twelve divided by four
that's three
x cube divided by x cubed that's one one
times three that's three only
y to the power five divided by y squared
as i said just subtract the exponent
five minus two that is
three so three y cubed
minus
okay minus 20 divided by four
that is five
okay x to the power of 5 divided by x
cubed
so just subtract the exponent that would
be
5 minus 3 that is squared
okay for y squared divided by y squared
is 1
multiplied by 5x squared obviously this
5x squared
then z since there is no z in the
denominator so just write it
okay so meaning
the factors for 12 x cubed y
to the power 5 minus 20
x to the power of 5
y squared z are
4 x squared i am in cube
y squared multiplied by three
y cubed minus
five x squared
z class
remember i told you you can check your
answers by multiplying the factors
and then you come up with the given
polynomial
okay another example so you have your 28
x cubed z squared minus
14 x
squared y cubed plus
36 y z to the power of four
so you have your three terms it's
trinomial
okay so just let's copy first
28
x cubed z cubed
minus 14 x squared
y cubed plus 36
y z to the power of four
so the same thing you're going to find
our gcm f
so let's start with the numbers 28 14
36 so for this class when you're going
to use the listing method or the
prime factorization you come up with
they just the gcmf of
okay 36 is 4 times nine fourteen is
seven times two twenty eight is
um seven is four so the
common is two
okay okay
okay i'll illustrate you birds
it's if you can
um visualize it okay let's have beer
28 14
36 so 28 prime factorization tile
fourth that's four times seven so four
is it's not prime so it's two
times two times seven
fourteen is two times seven
thirty-six is four times nine
yes nine four is two
times two nine is
the three times three so very clear
they're common factors form factor
it's just two that is why i write a well
ago
two okay
okay so there you see
for the numbers there you see f
or just mf is 2. now proceed with the
variables so x cubed x squared there is
no x for the third term
so meaning x is not gcf or just
math how about for y
for the second term is y cube third term
y but there is no y for the first term
so y is not included in the gcf
how about the z z cube for first term
z to the power of 4 for third term
so meaning z is not common to the three
terms
so only two is the gcm f
now proceed you have to find the other
factors so divide each
term by the gcmf so 28
x cubed z
cubed minus 14
x squared y cube
plus 36 y
z to the power of four divided by
[Music]
h by two
okay so
two times 28 divided by two
that is fourteen
i by the way class i divide two for the
three terms
you can also divide two by each term
that's the same answer
okay so capital x cubed
z cubed then
fourteen divided to the seven so minus 7
x squared y to the power of cube
36 divided by 2 that is
18 18 y
z to the power of four so
that is the final answer
let's have fifth example
so you have there
five h plus 40 h k cubed
okay let's check first if the two terms
are perfect cube suit
since there are two terms it could be
sum of two cubes but
check if they are perfect you so 5h that
is not a perfect cube 40h
is not also perfect cube so it might be
there is gcmf or greatest common
monomial factor among the two terms so
what would be the
gcmf
so the gcmf is 5h
okay you factor 5h for the two terms so
you divide each term 5h
divided by 5h plus
40 h
k cubed divided by 5h
so you have 5h 5h divided by 5h that is
1
plus 40 divided by 5
that is 8. h divided by 8 it's one also
eight times one is eight and then k
q okay
previously we had already
discussed the
factor of 1 plus 8 k cubed
factor for 1 plus 8 k cubed from our
example number 4
are the sum of 1 and two k
then one plus
two k plus
four k squared
that's all for sum and difference of
blue cubes stay tuned for another lesson
on factoring techniques god bless
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