Factor Polynomials - Understand In 10 min

TabletClass Math
13 Feb 201914:06
EducationalLearning
32 Likes 10 Comments

TLDRThis video script offers essential tips on factoring polynomials, crucial for success in algebra and related math courses. It introduces four common scenarios for factoring, emphasizing the importance of starting with the greatest common factor (GCF). For trinomials without a GCF, two cases are discussed: 'case one' with a leading coefficient of 1, and 'case two' with a different coefficient, using strategies like the 'double smiley face' technique. The script also touches on special factoring rules, such as the difference of squares. The instructor suggests additional resources for those seeking more in-depth instruction.

Takeaways
  • πŸ“š The video aims to provide tips on factoring polynomials, covering the most common situations encountered in algebra and algebra 2.
  • πŸ”‘ The presenter emphasizes that understanding multiplication of polynomials is crucial for successful factoring, suggesting that without this knowledge, passing algebra could be difficult.
  • πŸ“‰ The video outlines four scenarios that cover most polynomial factoring situations, suggesting a structured approach to tackling different types of polynomials.
  • 🌟 The first scenario is identifying and factoring out the Greatest Common Factor (GCF), which is the starting point for any polynomial factoring.
  • πŸ“ The script provides a brief tutorial on how to factor out the GCF, illustrating with examples and suggesting that viewers check additional resources for a deeper understanding.
  • πŸ” After checking for a GCF, the next step is to look for trinomials, specifically distinguishing between 'case one' trinomials with a leading coefficient of 1 and 'case two' trinomials with a different leading coefficient.
  • πŸ“ The video introduces a method for factoring 'case one' trinomials by finding pairs of factors of the constant term that add up to the middle term's coefficient.
  • πŸ˜€ For 'case two' trinomials, the presenter introduces the 'double smiley face' technique, which involves factoring the leading coefficient and finding factors of the constant term that lead to the correct middle term when combined.
  • πŸ“– Special factoring scenarios are mentioned as the last resort if neither a GCF nor a trinomial situation applies, with the 'difference of squares' being highlighted as a key special factoring rule.
  • πŸ“š The presenter wraps up by stating that the video provides a foundation but not a complete education on factoring, encouraging viewers to seek further instruction through additional videos or math courses.
  • πŸ‘ The video concludes with a call to action for viewers to subscribe, hit the notification bell, like the video, and leave comments for feedback, indicating the presenter's engagement with the audience.
Q & A
  • What is the purpose of the video?

    -The purpose of the video is to provide powerful tips on how to factor polynomials commonly encountered in algebra, algebra 2, or college-level math courses.

  • Why is it important to know how to multiply polynomials before learning to factor them?

    -It's important to know how to multiply polynomials because understanding the multiplication process helps in reversing the process, which is essentially what factoring is.

  • What is the first step you should take when attempting to factor a polynomial?

    -The first step is to check if you can factor out a Greatest Common Factor (GCF) from the polynomial.

  • What should you do if there is no Greatest Common Factor in a polynomial?

    -If there is no GCF, you should then check if the polynomial is a trinomial and determine its type.

  • How do you factor a trinomial where the leading coefficient is 1?

    -For a trinomial with a leading coefficient of 1, you list all pairs of factors of the constant term and find the pair that adds up to the middle coefficient.

  • What is the double smiley face technique?

    -The double smiley face technique involves multiplying certain pairs of factors to find the correct factorization of a trinomial where the leading coefficient is not 1.

  • What should you check if a polynomial is neither a GCF case nor a trinomial?

    -You should check if the polynomial fits any special factoring scenarios, such as the difference of squares.

  • What is the difference of squares formula?

    -The difference of squares formula is a^2 - b^2 = (a + b)(a - b).

  • What are the four main scenarios covered for factoring polynomials in this video?

    -The four main scenarios are: factoring out the GCF, factoring a trinomial with a leading coefficient of 1, factoring a trinomial with a leading coefficient other than 1, and using special factoring rules like the difference of squares.

  • What additional resources does the instructor suggest for those struggling with factoring polynomials?

    -The instructor suggests watching more of his YouTube videos or enrolling in his math courses for more extensive instruction.

Outlines
00:00
πŸ“š Introduction to Polynomial Factoring Techniques

This paragraph introduces the video's purpose, which is to provide powerful tips on factoring polynomials commonly encountered in algebra, algebra 2, or college math courses. The speaker emphasizes that while the video will be helpful, additional resources such as other videos on the YouTube channel or math courses are available for those who need more extensive instruction. The paragraph sets the stage for four different factoring scenarios that will be covered, starting with the importance of knowing how to multiply polynomials before attempting to factor them. It also highlights the necessity of understanding factoring for success in algebra-related classes.

05:04
πŸ” Identifying Polynomial Factoring Scenarios

The second paragraph delves into the process of identifying the different scenarios one might encounter when factoring polynomials. It outlines a systematic approach starting with looking for the Greatest Common Factor (GCF), then moving on to trinomials, and finally considering special factoring scenarios. The speaker introduces 'case one' and 'case two' trinomials, explaining that 'case one' involves a leading coefficient of 1, while 'case two' involves a different leading coefficient. The paragraph also introduces the concept of special factoring rules, such as the difference of two squares, and encourages viewers to check out additional resources for a deeper understanding.

10:06
πŸ“ Techniques for Factoring Trinomials and Special Cases

This paragraph focuses on the techniques for factoring trinomials, specifically 'case one' and 'case two', and special factoring scenarios. For 'case one' trinomials, the method involves finding pairs of factors of the constant term that add up to the linear coefficient. For 'case two', the 'double smiley face' technique is introduced, which involves factoring out the greatest common factor from the quadratic term and then finding factors of the constant term that, when combined with the linear term, yield the middle term. The paragraph also revisits the special factoring rule of the difference of two squares as an example of a special scenario. The speaker wraps up by emphasizing the importance of practice and provides a final reminder about additional resources available for those who need more help.

Mindmap
Double Smiley Face Technique
Factoring Technique
Difference of Squares
Case 2
Case 1
Application
Identification
Encouragement
Summary
Math Courses
YouTube Channel
Special Factoring Scenarios
Trinomials
Greatest Common Factor (GCF)
Multiplication Knowledge
Necessity
Audience
Purpose of the video
Conclusion
Additional Resources
Strategies for Factoring
Importance of Factoring
Introduction
Factoring Polynomials
Alert
Keywords
πŸ’‘Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial into a product of simpler polynomials or factors. It is a fundamental skill in algebra that is essential for solving various mathematical problems. In the video, the instructor emphasizes the importance of factoring as a necessary topic to master, especially for students struggling with algebra or algebra 2.
πŸ’‘Greatest Common Factor (GCF)
The Greatest Common Factor, or GCF, is the largest factor that two or more numbers share. In the context of the video, the GCF technique is the first approach to take when factoring polynomials. The instructor explains that if a polynomial has a common factor in all its terms, it should be factored out first, simplifying the expression.
πŸ’‘Trinomial
A trinomial is a polynomial that consists of three terms. In the video, the instructor discusses two types of trinomials: those with a leading coefficient of 1 (referred to as 'case one') and those with a leading coefficient other than 1 ('case two'). The method of factoring these trinomials is a key focus of the video.
πŸ’‘Case One
In the script, 'case one' refers to a specific type of trinomial where the leading coefficient is 1. The instructor provides a method to factor these trinomials by finding two numbers that multiply to the constant term and add up to the linear coefficient.
πŸ’‘Case Two
Similar to 'case one,' 'case two' in the video refers to another type of trinomial, but with a leading coefficient that is not 1. The instructor introduces a technique called the 'double smiley face' to factor these trinomials, which involves a slightly more complex process than 'case one'.
πŸ’‘Special Factoring Scenarios
Special factoring scenarios are unique situations where standard factoring techniques may not apply, and specific rules or patterns must be recognized to factor the polynomial correctly. An example given in the video is the difference of squares, which is a special case that can be factored using the formula a^2 - b^2 = (a + b)(a - b).
πŸ’‘Difference of Squares
The difference of squares is a special factoring rule that applies to expressions of the form a^2 - b^2. The video script mentions this as a key special factoring rule, where the expression can be factored into (a + b)(a - b), which is a fundamental concept in algebra.
πŸ’‘FOIL Method
The FOIL method is a technique used for multiplying two binomials. Although not explicitly detailed in the script, the instructor implies that understanding how to multiply polynomials, such as using the FOIL method, is a prerequisite to successfully factoring them.
πŸ’‘Distributive Property
The distributive property is a fundamental arithmetic operation that allows you to multiply a number by each element inside a parenthesis and then sum the results. The video script suggests that understanding this property is crucial for both multiplying and factoring polynomials.
πŸ’‘Algebra
Algebra is a branch of mathematics that uses symbols and the rules of operations to manipulate and solve equations. In the video, the instructor discusses factoring polynomials as a critical skill within algebra, particularly for students taking algebra or algebra 2 classes.
πŸ’‘Math Courses
The instructor mentions math courses as a resource for students who may require more extensive instruction on factoring polynomials. These courses are offered as an alternative for those who need a more formal and structured approach to learning the material presented in the video.
Highlights

The video aims to provide powerful tips on factoring polynomials, covering the most common situations in algebra, algebra 2, and college math.

The presenter emphasizes the importance of understanding polynomial multiplication as a prerequisite for successful factoring.

The video offers a structured approach to factoring polynomials, starting with identifying the greatest common factor (GCF).

A detailed explanation of how to factor out the GCF from a polynomial is provided, with an example to illustrate the process.

The presenter introduces four different scenarios for factoring polynomials, which cover the majority of situations encountered in algebra classes.

A special focus is placed on trinomials, with two specific cases (Case 1 and Case 2) discussed in detail for factoring.

Case 1 trinomials, with a leading coefficient of 1, are factored by finding pairs of factors that add up to the middle term's coefficient.

Case 2 trinomials, with a leading coefficient other than 1, are approached using the 'double smiley face' technique to find the correct factors.

The video explains the process of identifying and applying special factoring rules, such as the difference of squares, to polynomials.

The presenter provides a method to verify factored polynomials by multiplying the factors to ensure they match the original polynomial.

Additional resources, including more videos and math courses, are suggested for those who need extensive instruction on factoring polynomials.

The video encourages practice as a key to mastering polynomial factoring, especially for those struggling with the concept.

The importance of being able to factor polynomials is stressed as a necessity for passing algebra and related math classes.

The presenter offers a mental organization strategy for approaching polynomial factoring, starting with GCF, then trinomials, and finally special scenarios.

A call to action is made for viewers to subscribe to the presenter's YouTube channel and engage with the content through likes and comments.

The video concludes with a reminder of the importance of understanding polynomial factoring and the availability of further help through the presenter's courses.

Transcripts
00:00

okay how to factor polynomials so the

00:04

purpose of this video is I'm going to

00:07

just try to give you some powerful tips

00:09

on how to factor the most polynomial

00:12

situations you're gonna come across in

00:14

in algebra or algebra 2 or college

00:17

chapter but of course you might be that

00:19

involves factoring polynomials these are

00:22

gonna be the most common situations now

00:23

I just want to tell you right up front

00:25

that this is you know you're not gonna

00:27

be able to if you're lost in this

00:29

subject if you will this is going to

00:30

help you out but it's not gonna be

00:32

enough so I have additional videos on my

00:34

youtube channel but if you really need

00:36

extensive instruction and I kind of

00:38

suggest that's the case for a lot of

00:41

people who are struggling they might

00:42

want to check out my math courses I'll

00:44

leave a link in a description of this

00:46

video if you're interested in learning

00:48

more from me and kind of in a formal

00:51

manner but with that being said let's

00:54

get into these four different type of

00:56

problems that will cover the majority of

00:59

polynomial situations that you may face

01:01

in any one of these particular or math

01:03

classes that you might be taking now

01:05

before you can factor a polynomial you

01:08

have to make sure that you can multiply

01:10

polynomials so if you don't know how to

01:12

multiply two polynomials together like

01:14

using the foil method or the

01:15

distributive property then you're then

01:18

you're gonna really struggle factoring

01:20

polynomials and I'll say additionally if

01:23

you can't factor polynomials then you're

01:26

going to struggle the department's going

01:29

to be impossible for you to pass your

01:30

algebra class or whatever other math

01:32

class you might be doing if it involves

01:35

algebra so this is an absolutely

01:37

necessary

01:39

topic to to learn in math okay so I've

01:43

got four different situations here and

01:45

let's get into it and if you understand

01:47

these four scenarios then you're gonna

01:50

be able to really handle most polynomial

01:55

factoring problems that you encounter in

01:57

algebra alright so the first is this

02:00

problem and this represents the GCF

02:04

technique now the GCF is a greatest

02:06

common factor you always always start

02:09

when you're looking at a polynomial to

02:11

see if you can factor out a greatest

02:13

common

02:14

factor okay now here you can so each one

02:17

of these proms that I have done your are

02:20

fact able so if you want to pause the

02:21

video just factor them real quick then

02:23

out of you know then see my answers and

02:25

that's kind of a good little pop quiz

02:26

for you and these are pretty simple

02:28

proms some I wouldn't get in a silly too

02:33

overly confident if you can handle these

02:34

but that's a it's a good indication that

02:36

you know what you're doing anyways let's

02:38

get into this the greatest common factor

02:39

is your the first place you always start

02:43

when you see up on the walls seemed he

02:45

could factor out a greatest common

02:47

factor now here what that is is the

02:50

greatest well it's exactly what the name

02:53

says it's the greatest common factor so

02:55

what's common amongst these two terms

02:58

here in terms of a number well it's four

03:01

okay then they have the highest power of

03:05

X this is X cubed this is x squared but

03:08

they're old but they only share an x

03:10

squared there's there is in common so

03:14

you would factor out an x squared like

03:16

so and that would leave you with an X

03:19

right here minus 2 okay now if you

03:23

weren't sure if this was the correct

03:26

answer you can multiply these together

03:28

and you can see you'd get back to this

03:30

answer okay and so this right here for x

03:35

squared is the greatest common factor

03:37

okay that is the greatest common factor

03:39

now again in a short period of time I'm

03:43

not going to be able to teach you

03:46

everything you need to know about the

03:47

greatest common factor so you should be

03:49

somewhat familiar with how to do this

03:50

but when it comes to factoring

03:52

polynomials is this is the number one

03:54

place you want to start okay so if you

03:57

don't know how to if you're not

03:59

comfortable with what factored out the

04:01

greatest common factor I have videos on

04:03

my youtube channel I go into it plus you

04:07

know you might need more extensive

04:08

instruction so you might want to check

04:10

out one of my courses just check out the

04:11

link below okay so that's the first

04:13

scenario okay now the second scenario is

04:17

this if you can't factor out a greatest

04:19

common factor okay let's say you're

04:22

looking at a polynomial and there is no

04:24

greatest common factor well that doesn't

04:25

mean that you're done

04:27

okay what you may have is one of these 3

04:30

remaining situations okay so let's just

04:32

kind of talk about these here so this is

04:35

what we call a trinomial there's three

04:38

terms but there's no greatest common

04:40

factor this is also a trinomial there's

04:43

three terms but there's no greatest

04:45

common factor and the difference between

04:47

these two trinomials is this one is just

04:50

a 1 x squared there's just a 1 in front

04:53

of it and then this has a number other

04:55

than 1 so here in this example this is 2

04:58

okay so what we have here are trinomials

05:03

trinomials so this is where you want to

05:06

look for next ok so you checked out

05:10

greatest common factors and you're gonna

05:11

see if there's any trinomials so the

05:13

last problem here you might encounter is

05:17

a special factoring scenario okay so

05:21

just we'll put the word here special so

05:23

these four scenarios here four

05:26

situations will cover the majority of

05:28

your factoring scenarios ok now I'm

05:32

gonna get into these problems these last

05:33

three problems here in a second but I

05:34

just wanted to just lay out the kind of

05:40

like you're a mental organization in

05:42

terms of hey I got a factor a polynomial

05:43

I always start with the GCF if I'm

05:46

dealing with a trinomial what type is it

05:48

I like to refer to this as a case one

05:51

because there's a wonder front of it and

05:53

then anything else is what we call like

05:55

say a case two

05:56

all right and if it's if you don't have

05:58

a case one or case two then see if

06:01

there's any special factoring rules that

06:03

uh that could apply to the polynomial

06:06

okay so let's get into this so very

06:10

briefly the case one is a trinomial

06:15

where there's just a 1 in front of the

06:18

leading it so the 1 is the leading

06:20

coefficient okay so when you write it in

06:22

standard form so if you look here the

06:25

easiest way to factor a case 1 if if

06:28

they are factorable okay let me just do

06:31

it this way is look at this last number

06:34

okay

06:35

that's negative 6 now one way I kind of

06:39

like to start students off

06:42

a factor in case once is to write out

06:44

all the factors of a negative six so

06:48

here's how you do okay so negative six

06:50

you can write as one times six a

06:52

negative one times six right will give

06:55

you a negative six 1 times a negative

06:58

six will give you a negative six okay

07:01

two times three negative two times three

07:05

and two times the negative three so

07:07

these are all the different ways you can

07:09

write the factors of negative six right

07:14

all these numbers these different

07:16

combinations will when you can multiply

07:17

these these pairs together we'll get you

07:19

a negative six now if you add up each

07:22

one of these pairs which what do you get

07:24

here you get a positive five right

07:26

negative one plus six is a positive 5

07:28

one plus negative 6 is negative five

07:32

this right here is a one and this is a

07:35

negative one right so when you add all

07:37

these pairs together so what you want to

07:40

do is to see if you have a pair any

07:43

pairs of factors that add up to this

07:45

Center number okay so this is a one x

07:49

squared this is a positive one so which

07:52

one of these pairs adds up to a positive

07:54

one it's these pairs right negative two

07:57

plus three gives you a positive one and

07:59

these are the answers these are the

08:01

factors so you could write this

08:03

trinomial you can factor it this way

08:06

okay you're always going to have two

08:08

binomials so it's going to be X minus

08:11

two

08:12

okay one of those answers and X plus

08:15

three right there so these are the

08:19

factors to that trinomial as simple as

08:22

that

08:23

okay of course you need to practice this

08:25

now if you couldn't find any pairs of

08:29

factors here that add up to that Center

08:31

number then a factor and then the

08:34

polynomials unfactored

08:37

so this one one in this case two now the

08:40

case two you could do you can do this in

08:42

a similar manner but there's some

08:44

additional steps but I'm gonna give you

08:46

another technique you can use to try to

08:49

factor a case to let me write this a

08:51

little better I kind of call it the

08:54

we'll smiley face so the first thing is

08:57

we have 2x squared so we want to write

09:00

the factors of 2x squared doesn't

09:02

there's only one way to factor that that

09:04

would be a 2x and X right so if I

09:07

multiply 2x and X together I get a 2x

09:10

squared there's no other way I could

09:12

write that now what we're trying to do

09:15

here is play a little game to try to get

09:19

back to this Center number now the way

09:21

we have to do that is write the factors

09:23

of negative 5 in this position right

09:27

here you'll see how this comes together

09:28

here in a second so negative 5 is what 1

09:32

times negative 5 or negative 5 times 1

09:34

so I'm gonna put this negative 5 right

09:37

there and I'm gonna put a 1 right there

09:39

okay now if what if what I'm gonna show

09:44

you doesn't work I can just kind of

09:45

maneuver these combinations around

09:47

because I'm trying to get back to the

09:50

center number now I told you I was gonna

09:52

use something called a double smiley

09:54

face technique so what that is is you

09:57

take this number here and you're

09:59

multiply it by this okay and that's one

10:01

smile when you face if you will and then

10:03

we do this times this so you can see we

10:05

have two smiley faces let me draw this

10:08

the other one a little bit bigger okay

10:11

so we have one X right 1 times X is X or

10:15

1 times X is 1 X positive 1 X and then I

10:18

have 2 x times a negative 5 that's

10:20

negative 10 X now if I add these two

10:24

guys together you're looking to see

10:27

which combination gets back to the

10:29

center term so if you see here a 1 X a

10:33

positive 1 X plus a negative 10 X gives

10:35

me a negative 9 X so this these factors

10:41

right here are correct okay because of

10:43

that and if you wanted to you can just

10:45

multiply all this out to verify that in

10:47

fact you have the correct factors okay

10:49

so this is the double smiley face now

10:52

you could do this problem and they're

10:55

similar fashion as I did this first case

10:57

one problem but there's some additional

11:00

steps so I really like to kind of the

11:02

double smiley face technique for this

11:04

case two polynomials and for case one

11:07

this this technique I

11:08

showed you here is just really

11:09

straightforward okay so I would suggest

11:10

using that as well okay so now you have

11:16

three things we have three things behind

11:19

us right we have the greatest common

11:20

factor we always look there first then

11:23

if we're dealing with trinomials what

11:24

type and if we don't have any trinomials

11:27

we don't have any greatest common factor

11:28

that doesn't mean that you're done you

11:30

may have a special factoring scenario so

11:32

if you look here this last problem

11:34

there's no GCF it's not a trinomial so

11:38

what do you do well in this case you

11:39

just need to know that special factoring

11:41

rules probably one of the most important

11:44

is a squared minus B squared owes me

11:47

write this this way here a squared minus

11:51

B squared is equal to a plus B times a

11:57

minus B it's called the difference of

11:58

two squares it's used extensively okay

12:02

when we're talking about factoring so

12:04

here the way I would factor this I would

12:07

just simply need to know this rule this

12:09

is a special factoring role so this is

12:11

going to be X plus 3 times X minus 3 ok

12:16

all right then a little bit better so

12:20

this is how this factors here because I

12:22

know the special factor in a rule now

12:24

again I'm gonna wrap up this video here

12:27

no way I could I fit in what takes your

12:31

teacher a couple weeks to teach you at

12:36

least the factoring is it's covered you

12:39

know and over and over a multiple math

12:41

courses so this is this is a lot of you

12:44

know skills that kind of build up no way

12:46

I should you expect to just watch this

12:48

video be like a total expert but if you

12:50

have a clue about factoring and you know

12:52

how to multiply and you're like I just

12:54

kind of struggling are a little bit

12:55

confused then I think this video would

12:57

definitely should have helped you out

13:00

right because you're always start here

13:01

you'll always start with the GCF then

13:04

look to see if you have any of these

13:05

scenarios and then if you don't just

13:07

make sure you don't have any special

13:08

scenarios if you follow this you're

13:09

going to be good to go in most factoring

13:12

situations but again if you need more

13:14

help you want to go ahead and check out

13:18

some additional videos I have on my

13:19

youtube channel so please consider

13:21

subscribing and if you do hit that Bell

13:24

notification and if you liked this video

13:27

hey you know give me a thumbs up I

13:29

appreciate that

13:29

and leave me comments it's um it's one

13:32

of the things that I tried to read it I

13:34

do get a lot of comments on my videos

13:36

which I'm grateful for but it gives me

13:37

feedback on on videos that I future

13:40

videos I can make that are going to help

13:41

help you out again

13:44

you know if you like my teachings down

13:46

you understand you know you know how I

13:50

teach and you need extensive help in

13:52

math then you might want to consider

13:54

check it out some of my math courses and

13:57

I'll leave a link in a description in

13:59

the video if you're interested there but

14:01

other than that I appreciate your time

14:03

and have a great day