Trinomials (M2 2.3 Lesson)
Summary
TLDRThe script is a tutorial on factoring polynomials, emphasizing the importance of understanding the underlying principles before using shortcuts. It introduces a method for factoring trinomials by finding two numbers that multiply to the product of the leading coefficient and constant term, and add up to the linear coefficient. The process involves splitting the middle term and using factoring by grouping. The script also covers factoring when the leading and constant terms are perfect squares and extends the technique to expressions with higher powers. It concludes with a problem-solving example, illustrating how to apply the method to solve equations.
Takeaways
- 📚 The video script is a tutorial on factoring polynomials, emphasizing the importance of understanding the underlying principles before using shortcuts.
- 🔑 The first method discussed is factoring out the Greatest Common Factor (GCF), which involves identifying a common factor in all terms of a polynomial.
- 🔍 The second method involves a strategic approach to factoring trinomials with three terms, where the script introduces a game-like method to find two numbers that meet specific multiplication and addition criteria.
- 📈 The tutorial explains how to use the coefficients of the polynomial to guide the factoring process, particularly focusing on the lead coefficient, constant term, and linear coefficient.
- 🎯 The script provides a step-by-step guide on how to split the middle term of a trinomial based on the two numbers identified, which is crucial for the factoring process.
- 🔄 The concept of 'factoring by grouping' is introduced as a method to simplify polynomials once they are expanded to four terms through strategic splitting.
- 📝 The tutorial emphasizes the importance of checking work by multiplying the factored form back to its original polynomial form to ensure accuracy.
- 🧩 The script discusses strategies for dealing with larger numbers and perfect squares in polynomials, suggesting tricks like prime factorization and recognizing patterns.
- 🔢 The video also touches on factoring polynomials that are not quadratic, such as those with terms raised to the sixth power, showing adaptability in the factoring techniques.
- 📉 The tutorial concludes with a problem-solving example, demonstrating how to apply the learned factoring methods to solve equations using the zero product property.
Q & A
What is the first type of factoring mentioned in the script?
-The first type of factoring mentioned is factoring out the Greatest Common Factor (GCF).
Why is it important to understand why shortcuts work before using them?
-It's important to understand why shortcuts work to ensure you're using them correctly and to apply them effectively in more complex problems.
What is the second type of factoring discussed in the script?
-The second type of factoring discussed is factoring by grouping, which is typically applied when there are an even number of terms.
Why can't the method of factoring by grouping be used for three terms?
-Factoring by grouping cannot be used for three terms because it requires an even number of terms to be split into pairs.
How does the script suggest finding two numbers for factoring when the terms are not easily divisible?
-The script suggests using prime factorization and grouping factors in sets of two to find two numbers that multiply to a given product and add to a given sum.
What is the significance of the numbers 18 and 2 in the script?
-The numbers 18 and 2 are found to be the two numbers whose product is 36 (3 times 12) and whose sum is 20, which allows for the successful factoring of the given expression.
How does the script recommend checking the factoring work?
-The script recommends checking the factoring work by multiplying the factored terms and comparing them to the original expression.
What is the strategy for factoring when the numbers are large or not easily factorable?
-The strategy is to look for patterns, such as perfect squares, and use prime factorization to break down the numbers into manageable parts.
Why is it important to match the terms when splitting up the middle term during factoring?
-It's important to match the terms to ensure that the factoring process is mathematically correct and that the terms can be grouped properly for further factoring.
What does the script suggest for factoring expressions that are not quadratic?
-The script suggests that the same factoring techniques can be applied to non-quadratic expressions by making necessary adjustments, such as ensuring the terms match when splitting.
How does the script demonstrate solving a problem using factoring?
-The script demonstrates solving a problem by first setting the equation to zero, then factoring, and finally using the zero product property to find the solutions.
Outlines
📚 Introduction to Factoring Techniques
The speaker introduces the concept of factoring in mathematics, emphasizing the importance of understanding the underlying principles before resorting to shortcuts. They differentiate their teaching method from traditional approaches and focus on explaining why certain factoring methods work. The first method discussed is factoring out the greatest common factor (GCF), which involves identifying a common factor among all terms in a polynomial. The speaker provides a step-by-step guide on how to factor out the GCF, starting with a three-term polynomial and explaining the process of identifying the GCF and factoring it out.
🔍 Factoring by Grouping and Prime Factorization
In this segment, the speaker explores the technique of factoring by grouping, particularly useful when dealing with polynomials that don't have an even number of terms. They demonstrate how to split terms to create pairs that can be factored, using the example of a polynomial with three terms. The speaker also introduces the concept of prime factorization as a strategy to find two numbers that multiply to a given product and add up to a specific sum, which is crucial for the factoring process. They guide through the process of breaking down a number into its prime factors and then grouping them to find the necessary numbers for factoring.
🔢 Factoring with Perfect Squares and Large Numbers
The speaker discusses a specific trick for factoring when the first and last coefficients of a polynomial are perfect squares. They explain how this observation can simplify the factoring process and provide a shortcut. The segment includes a detailed example where the speaker guides through the process of factoring a polynomial with large coefficients, emphasizing the importance of recognizing patterns and using prime factorization to assist in the factoring process. The speaker also highlights the need to adjust the sign of terms to ensure the correct sum and product when factoring.
🔄 Adjusting Factoring Methods for Non-Quadratic Polynomials
This paragraph delves into factoring techniques for polynomials that are not quadratic but still have three terms. The speaker explains how to adapt the factoring by grouping method for these cases, with a focus on ensuring that the terms being split are of the same kind as the middle term in the original polynomial. The segment includes an example of a polynomial with terms raised to the sixth power, demonstrating how to find the correct pair of numbers to multiply and add to the desired coefficients, and then how to split and factor the polynomial accordingly.
🧩 Solving Equations Using Factoring
The final paragraph shifts focus to solving equations using the zero product property and factoring. The speaker guides through the process of transforming an equation to set it equal to zero, which is a prerequisite for applying the zero product property. They demonstrate how to factor the resulting polynomial and then solve for the variable by setting each factor equal to zero. The speaker concludes with an interactive problem-solving example, engaging the audience to participate in the factoring process and solve the equation together.
Mindmap
Keywords
💡Factoring
💡Greatest Common Factor (GCF)
💡Factor by Grouping
💡Lead Coefficient
💡Constant Coefficient
💡Linear Coefficient
💡Perfect Squares
💡Zero Product Property
💡Difference of Two Perfect Cubes
💡Prime Factorization
Highlights
Introduction to a new method of factoring that differs from traditional approaches.
Emphasis on understanding the logic behind mathematical shortcuts before applying them.
Explanation of the first type of factoring: factoring out the Greatest Common Factor (GCF).
Discussion on why the second type of factoring, factoring by grouping, is not applicable in this case due to an odd number of terms.
Introduction of a third type of factoring method suitable for trinomials.
A game-like approach to finding two numbers that multiply to a given product and add up to a given sum.
Step-by-step guide on how to split the middle term of a trinomial based on the two numbers found.
Transformation of a trinomial into a binomial through factoring by grouping.
The importance of checking factored solutions by multiplying them out to ensure accuracy.
A strategy for factoring when dealing with larger numbers by using prime factorization.
An example of how to group prime factors to find two numbers that meet specific multiplication and addition criteria.
The use of the zero product property in solving quadratic equations.
Application of the new factoring method to a problem with a negative leading coefficient.
A special trick for factoring when the first and last coefficients are perfect squares.
Adjusting the factoring method for expressions with terms raised to the sixth power.
Final thoughts on when to stop factoring and the limitations of the method being taught.
Invitation for students to choose a problem to solve together, highlighting interactive learning.
Real-time demonstration of solving a quadratic equation using the factoring method chosen by the students.
Transcripts
okay so you guys may have had some
experience factoring how many of that
experience factoring remember doing in
the past okay I must do let us don't and
you may the way I'm teaching you today
is going to be different than the way
you probably learned it's going to be
remember
there's in math or shortcuts and
shortcuts are good because they're
quicker to get especially when you have
really complicated problems and a lot of
stuff to do you want to take shortcuts
but before you take the shortcuts you
want to slow down and make sure you
understand why do they work and so right
now we're we are doing factoring the way
that explains why it works and
eventually we'll get to problems where
factorings not the focus but we'll need
to factor and we'll probably start
taking some shortcuts okay so I want you
for right now even even if you know how
to do these a different way I want you
to use it this way okay so the first
type of factoring is to factor out a GCF
so you look at all three terms we're
going to do number six together right
now is there something that all three
terms have in common that we can factor
out other than one or itself yeah I know
then second type of factoring is what we
did yesterday if you have four terms or
an even even a like six terms aren't
even number where you can split it in
half and we don't have that we got three
terms odd number right so you can't
factor by grouping so now we need a
third type of factoring and so that's
what we're looking at here so the first
thing you do is play a little game and
I'm going to draw an X here and I'm
going to ask myself what two numbers
multiply to 3 times 12 which is 36 and
add to 20 now let me show you where
things are coming from the 3 and the 12
are your lead
and your constant coefficients your
twenty is your linear coefficient okay
so you're using all three of those
numbers so you want to multiply the
first and the last number together and
you want to just think about that second
number so can we come up with two
numbers whose product is 36 and whose
sum is 2030 18 and 2 alright so 18 times
2 is 36 18 plus 2 is 20 right does it
work so if it works
then that tells you that this methods
gonna factor if this doesn't work that
tells you it's not going to factor using
this method okay so this is kind of once
you get this far then you know
everything's gonna work out fine so once
you have the 18 and the 2 that's going
to split up your middle term right now I
have 20 DS I'm going to split them up to
18 DS plus 2 DS so it still adds up to
20 by splitting it up in a specific way
coming from the 18 and the two that we
found here so 3 d squared plus 12 I'm
going to bring my first my last term
down video gave that now what we
created four terms instead of three
terms not anybody have an idea what the
deal yeah do what we did last night so
now you can software by grouping so
these two terms factor out a 3d what's
left what do you get when you factor out
turns D plus six plus factor out of two
it was left D plus six and now both
terms have a common factor of D plus six
so factor out the D plus six and you're
left with three D plus two that's our
answer
and we're asked to check it how we gonna
check this yeah so I split the 22 18
into so fine I'm well it's a common term
what is 18 D plus 2 D is 20 so so I have
to I have to me I have to keep the DS
there right I'm just splitting them up
but yes that's where they're coming from
so for when you're checking you've got
to take your answer D plus 6 and 3 D
plus 2 and multiply that out and go
ahead and multiply that out and check
your answer
so to work out okay so compare your your
factoring work with your checking work
and notice that the last line over here
is the first line over here the second
to the last line on your factoring is
exactly the second line here right or
the third line time the second line here
is the third line they're the first
lines last night so it's exactly reverse
order
okay so let's try it again a number
eight I'm going to skip some what I skip
is homework for you okay so eight we're
going to start by seeing if it factors
we're looking for two numbers that
multiply to Wed 3 times negative 40
what's 3 times negative 40 so it's got
to multiply to negative 120 and it's got
to add to negative seven let's see if
you could figure those out
how many found this problem a little
harder to come up with in the last one
okay so eventually I'll you'll you'll
encounter a problem that's just like I
don't know what to do things mock like
this and add to this and it's nice to
have a little strategy to figure out
what two numbers multiply to 120
obviously it's 12 and 10 because you got
a zero at the end it's even so 2 times 2
times 60
good work what else what else multiplies
to 120 and so if you're if you're just
having trouble coming up with things on
the top of your head it's not so obvious
what what I do is I take to 120 I ignore
Nate the negative right now and I break
that number down to Prime's so 120 for
example it was 12 times 10 12 is 4 and 3
10 is 2 and 5 4 is 2 and 2 and so I know
120 is 2 times 2 times 2 times 3 times 5
those are the prime numbers that make up
120 and also make up all the factors
that multiply to 120 so if I group these
factors in sets of 2 I will get all that
I want so for example I can group that
way if I grouped that way then what's 2
times 2 times 2 times 3 so that's 24
times 5 all right so that's an option 24
times 5 isn't is 120 so I could use this
too and you try it out is there any way
to make 7 from 24 and 5 if I subtract
them or add them no so then I okay let's
try grouping in a different way I got
three twos a three and five let's try
this way well what's the three twos a
and 15 well if I add those I don't get
seven if I subtract into I get seven yes
so eight times 15 is what we want to
work with now now we want to think about
what should be positive which should be
negative right I know I want a 15 and
eight but which one should be negative
15 right because you have your answer is
negative 7 when you add them so you want
the bigger number to be negative so that
when you add and we get to negative
seven okay so so this this trick of
getting your prime factorization and
then just grouping them into two groups
and you could rearrange them so you
could also do something like this to put
the two at the five and then I got two
more twos and a three and that would
give me what 10 times 12 and we know
that one right but you can you don't
have to keep them in order from lowest
to highest you could rearrange them to
come up with okay so that's a good thing
to know especially when you're
struggling coming up with the two
numbers all right so we found them so
then the next step is to split this up
to negative 15 P plus 8 P and then we
carry down the 3p squared and the
negative 40 so we expanded it from three
terms to four terms now we can use our
factoring by grouping so what do we
factor out the first two terms 3p what's
left P minus five good and then last two
terms we could factor out a eight and
we're left with another P minus five and
then both of these have the same factor
P minus 5 so you factor that out
oops so P minus five times 3p plus eight
and then again
check that and all these the checking
for you for homework okay so let's try
number ten now you've never ten a chance
numbers are getting bigger I'll give you
a 30-second head start so when when the
numbers are really big look for one
thing here because sometimes this isn't
this is a trick that you want to look
for cuz it could happen and it does
happen quite a bit look at the first
number in the last number and check if
there will anybody notice what they are
what kind of numbers they're they're
both perfect squares so if you notice
the first number and the last number of
perfect squares almost always this trick
will work not always but almost always
and so you start the same way and saying
look at what two numbers multiply to
four times 81 and I'm not gonna even
multiply the madam's going to leave four
times 81 and add to negative 36 well 4
times 81 are both perfect squares so 4
times 81 4 is 2 times 2 81 is 9 times 9
and so if you pair them up you know you
kind of split
so I take one of the twos with one of
the nines and not the other two with the
other night what is that what product
will that give us what times wet
18:18 and just check if I add those do I
get 36 and usually it works like that if
you have perfect squares that's that's a
trick that you even use and doesn't
always work but if it does check that
one first because it'll save you some
time so 18 and 18 is what I want but I
can't
something wrong still how do I fix this
yeah you got to make them both negative
because the product is positive so I
need a negative times a negative because
when this the sums got to be negative
but the products got to be positive so I
know both terms are going to be negative
so then you split them up negative 18x
and another negative 18x bring down to
81
bring down the 4x squared factor by
grouping so the first two terms I can
pull out a 2x left with 2x minus 9 last
two terms I could pull out a negative 9
left with 2x minus 9 again both terms
have a 2x minus 9 and we have
another 2x minus 9 and so we have
matching factors here so if you have two
factors that match what's it what's a
shorthand for writing that yet that's
the same thing as 2x minus 9 quantity
squared and that would be our answer and
again you're going to want to check that
and I'll let you took that for homework
you check by doing what what which is
what's the opposite of factoring
multiplying good ok
now let's scroll down to the very number
1515 looks a bit different right it's
something to the sixth power not to the
2nd power so it's not quadratic it's the
power of six
and but our factoring techniques are not
limited to quadratics only and so number
14 and 15 are not quadratic but there's
still three terms and so we could try
our method on this we just have to
change a couple things about the method
so let's do fifteen see how things me
have to be adjusted so we still start
saying what two numbers multiply to what
1 times negative 32 which is negative 32
and add to what negative 4 so find those
two
numbers however you'd like to find them
let me know nu heaven negative eight and
four you agree Lindsey okay so now when
we split this up what am I going to
write here negative eight what R to the
third plus four what R to the third so
what's important here is that these two
are always like terms with this middle
term so this is an R to the third term
so both of these have to match it and if
you go back and look the two middle
terms always match that that middle one
in the trinomial and then we bring down
the 32 negative 32 we bring down the art
of the six and then factor by grouping
and everything else is going to work out
perfectly so it's just that that
adjustment just so you know that when
you split those up they have to be the
same kind of term as the term that's
being split up so when we factor this
what can we factor out R to the third
and we're left with R to the third minus
8 plus second one we could factor out a
4 we've got an R to the third minus 8
and then go to factor out the arch of
the third minus 8 and we're left with R
to the third plus 4 and at this point
that's where you would stop but the arts
of the third minus 8 will factor more
once you know how to factor a difference
of two perfect cubes
I don't looked at that yet this one
won't factor anymore and a lot to learn
why later on but this is as far as I'd
want you to go right now and so then you
can check that okay so those are
some new kind of factoring it it's
really how to split something up from
three to four terms so that we can use
factoring by grouping let's look let's
do a solving problem together I'll let
you pick one that we will do together
sixteen seventeen or eighteen your pick
seventeen okay I heard seventeen first
let's do seventeen but you tell me
seventeen what do we have to do first
why okay we want to make the
to zero so we're gonna have to subtract
30 from both sides of that equation and
we get C squared plus C minus 30 equals
zero because we want to use the zero
product property so it's got to be equal
to zero okay then we factor looking for
two numbers that multiply to one times
negative 30 so that's negative 30 and
adds to one what is it six and negative
five so we split it up C squared plus
six C minus five C minus 30 equals zero
and then we continue what we've learned
so we factor by grouping all right so
now it's factored I wish I had a little
more space but nowadays factored what
are we do we have our
times RB equals zero which means what
one of them has to equal zero so you
split your factors up get over here
so either C plus 6 equals 0 or C minus 5
equals 0 if C plus 6 is 0 then C is
negative 6 if C minus 5 is 0 then C is
positive 5 stop there
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