Trinomials (M2 2.3 Lesson)

Summary

The 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

00:00

📚 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.

05:19

🔍 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.

10:20

🔢 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.

15:20

🔄 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.

20:23

🧩 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.

Keywords

💡Factoring

Factoring is the process of breaking down a polynomial into a product of other polynomials or factors. In the video, factoring is the central theme, with the instructor demonstrating various methods to factor expressions, particularly focusing on polynomials with three terms. The video aims to teach viewers how to factor expressions by first finding the greatest common factor (GCF) and then using techniques like factoring by grouping.

💡Greatest Common Factor (GCF)

The GCF is the largest factor that two or more numbers share. In the context of the video, the GCF is the first method introduced for factoring polynomials. The instructor explains that one should look for a common factor that can be taken out from all terms of a polynomial, which simplifies the expression. For instance, in the example '3d^2 + 12d', the GCF '3d' is factored out.

💡Factor by Grouping

Factor by grouping is a technique used when a polynomial has more than three terms. The method involves grouping terms and factoring out the GCF from each group. The video demonstrates this by first transforming a three-term polynomial into a four-term polynomial by splitting the middle term, and then factoring out the common terms. This technique is showcased when the instructor works through the polynomial '3d^2 + 20d + 12'.

💡Lead Coefficient

The lead coefficient is the numerical factor that multiplies the highest power of the variable in a term. In the video, the instructor uses the lead coefficients, such as '3' in '3d^2', to help determine the two numbers that multiply to a certain product and add up to a specific sum, which is part of the factoring process.

💡Constant Coefficient

The constant coefficient is the numerical factor that stands alone in a term without being multiplied by any variable. In the video script, the constant coefficient is used in the process of finding two numbers that multiply to the product of the lead and constant coefficients and add up to the linear coefficient.

💡Linear Coefficient

The linear coefficient is the numerical factor that multiplies the variable to the first power in a term. The video uses the linear coefficient, such as '20' in '20d', to guide the process of finding the two numbers that will be used to split the middle term during factoring by grouping.

💡Perfect Squares

Perfect squares are numbers that are the product of some integer multiplied by itself. In the video, the instructor points out that if the first and last terms of a polynomial are perfect squares, a particular factoring trick can be applied. This is demonstrated when factoring a polynomial where the terms '4x^2' and '81' are both perfect squares.

💡Zero Product Property

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. In the video, this property is used to solve equations by setting a factored expression equal to zero and then solving for the variable.

💡Difference of Two Perfect Cubes

This is a special factoring formula that applies to expressions that are the difference between two cubes. The video hints at this concept when discussing the factoring of 'a^6 - b^6', which can be further factored using the difference of two perfect cubes formula, although the detailed method is mentioned to be covered later.

💡Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors, which are the prime numbers that multiply together to give the original number. In the video, the instructor uses prime factorization to help find the two numbers that multiply to a certain product and add up to a specific sum, which is crucial for factoring by grouping.

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