Factoring Tough Trinomials Using the Box Method

Summary

This video introduces the 'Box Method' for factoring trinomials, an alternative to the AC method. It simplifies the process by eliminating the need for factor by grouping. The method involves setting up a box with the quadratic term in the upper left, the constant in the lower right, and the middle terms derived from the product of 'a' and 'c' and the sum of 'b'. The video demonstrates the method with examples, showing how to find the middle terms and factor the trinomial into binomials. It's particularly useful for trinomials where 'a' is not 1, offering a clear and straightforward approach to factorization.

Takeaways

• 📚 The video introduces the 'Box Method' for factoring trinomials, which is a variation of the AC method.
• 🔍 The Box Method simplifies the process by eliminating the need for factor by grouping, which can be confusing for some.
• 📐 The method begins by writing the trinomial in descending order of powers, ensuring the equation is set up correctly.
• 🔢 The 'a' and 'c' values from the trinomial are multiplied to get the product, AC, and then two numbers are found that multiply to AC and add up to B.
• 📈 A box is set up with 'a' in the upper left, the constant 'c' in the lower right, and the two numbers found for the B value in the other two corners, each with an 'x'.
• 🔄 The trinomial's middle term is split into two terms that correspond to the numbers in the box, ensuring each has an 'x'.
• 🧩 The greatest common factor (GCF) of each row in the box is determined, which will correspond to the factors outside the box.
• 🔄 The GCFs from the rows are used to find the factors on the sides of the box, which when multiplied together, give the original trinomial.
• 🔍 The method is demonstrated through examples, showing how to factor trinomials like 9x^2 - 21x - 8 and 6x^2 + 13x - 5.
• 📝 The video emphasizes that the Box Method is particularly useful when 'a' is not 1, as it simplifies the factoring process for more complex trinomials.

Q & A

What is the primary focus of the video?

The video focuses on teaching a method for factoring trinomials called the 'Box Method,' which is a variation of the AC method.

Why might the Box Method be preferred over the AC method?

The Box Method might be preferred because it eliminates the factor by grouping step, which can be confusing for some people.

What is the first step in factoring a trinomial using the Box Method?

The first step is to write the trinomial in descending order of powers.

What are the roles of 'a', 'b', and 'c' in the context of the Box Method?

In the Box Method, 'a' is the coefficient of the x squared term, 'b' is the coefficient of the x term, and 'c' is the constant term.

How does the Box Method simplify the process of finding the two numbers that multiply to 'ac' and add up to 'b'?

The Box Method simplifies this process by visually organizing the numbers in a box, which helps in identifying the two numbers more intuitively.

What is the significance of placing the x squared term and the constant term in specific corners of the box?

The x squared term is placed in the upper left corner, and the constant term in the lower right corner of the box to set up the structure for factoring.

Why is it important to include 'x' in the two middle boxes of the Box Method?

Including 'x' in the two middle boxes is crucial because it represents the terms that will be factored out from the trinomial.

How does the Box Method ensure that the factors are correctly identified?

The Box Method ensures correct factor identification by using the greatest common factor of each row and checking that the products of these factors match the original trinomial.

What is the purpose of the greatest common factor (GCF) in the Box Method?

The GCF is used to simplify each row of the box, making it easier to identify the binomial factors that will multiply to give the original trinomial.

How can one verify that the factors obtained using the Box Method are correct?

One can verify the factors by using the FOIL method (First, Outer, Inner, Last) to multiply the binomials and check if they result in the original trinomial.

Is the Box Method suitable for all types of trinomials, or are there specific conditions for its use?

The Box Method is suitable for most trinomials, but it is particularly effective when the coefficient 'a' of the x squared term is not 1, as it simplifies the process of finding the correct factors.

Outlines

00:00

📚 Introduction to the Box Method for Factoring Trinomials

This paragraph introduces an alternative method for factoring trinomials known as the Box method, which simplifies the factor by grouping step that can be confusing. The example trinomial given is 9x^2 - 21x - 8. The process begins by arranging the trinomial in descending order of powers, resulting in 9x^2 - 21x - 8. The method involves identifying the coefficients a, b, and c, where a is the coefficient of the x^2 term, b is the coefficient of the x term, and c is the constant. The next step is to multiply a and c and find two numbers that multiply to this product and add up to b. For the example, a is 9, b is -21, and c is -8, leading to the calculation of a*c = -72. The challenge is to find two numbers that multiply to -72 and add to -21, which are identified as -24 and 3. These numbers are then placed in a box formation, with the x^2 term in the upper left and the constant term in the lower right, followed by the two numbers found, each with an x term.

05:01

🔍 Applying the Box Method to Factor a Trinomial

The paragraph demonstrates the application of the Box method using the trinomial 9x^2 - 21x - 8. The method involves splitting the middle term (-21x) into two terms (-24x and 3x) that correspond to the numbers found in the previous step. The next step is to factor by finding the greatest common factor (GCF) of each row in the box. For the top row, the GCF of 9x^2 and 3x is 3x, which is placed outside the box. For the bottom row, the GCF of -24x and -8 is -8, which is also placed outside the box. The resulting binomials are then multiplied to check the original trinomial, confirming the factorization. The paragraph emphasizes the importance of placing x terms in the box and using the GCF to determine the factors outside the box.

10:01

📘 Verifying the Box Method with an Additional Example

The paragraph presents a second example to further illustrate the Box method, using the trinomial 6x^2 + 13x - 5. The process begins with the AC method to find the two numbers that multiply to 6*(-5) = -30 and add up to 13, which are identified as 15 and -2. These numbers are then placed in the box, with the x^2 term in the upper left and the constant term in the lower right. The GCF for each row is determined, resulting in 2x and -1 for the top row, and 5 and 3 for the bottom row. The binomials formed by these factors are then multiplied to confirm the original trinomial. The paragraph concludes by highlighting the effectiveness of the Box method, particularly when the coefficient 'a' is not 1, and suggests that a different method might be easier for trinomials with 1x^2.

Keywords

💡Factoring

Factoring is the process of breaking down a polynomial into a product of other polynomials or factors. In the context of the video, the presenter introduces a method for factoring trinomials, which are polynomials of degree two with three terms. The video's theme revolves around simplifying this process, making it more accessible for those who find traditional methods challenging.

💡Trinomials

A trinomial is a polynomial with three terms. The video specifically discusses methods for factoring trinomials, which is a common algebraic task. The example given in the script, '9x squared minus 21x minus eight,' is a trinomial that the presenter aims to factor using the Box method.

💡Box Method

The Box method is a variation of the AC method for factoring trinomials. It simplifies the process by eliminating the need for factor by grouping, which can be confusing. The video demonstrates how to use the Box method to factor trinomials by setting up a 'box' or rectangle with the coefficients and solving for the factors that fit into this structure.

💡AC Method

The AC method is a technique used to factor trinomials where 'a' is the coefficient of the squared term and 'C' is the constant term. The method involves finding two numbers that multiply to 'a * C' and add up to the coefficient of the linear term ('B'). The video uses the AC method as a stepping stone to introduce the Box method.

💡Coefficient

A coefficient is a numerical factor multiplying a variable in an algebraic expression. In the video, coefficients are crucial for the factoring process. For instance, the coefficients 'a' (9), 'B' (-21), and 'C' (-8) are used to determine the factors needed for the Box method.

💡Greatest Common Factor (GCF)

The Greatest Common Factor is the largest factor that two or more numbers share. In the context of the video, finding the GCF is a step in the Box method to simplify the polynomial expression. The presenter uses the GCF to break down the polynomial into binomials that can be multiplied to yield the original trinomial.

💡Distributive Property

The Distributive Property is a fundamental algebraic principle that allows for the multiplication of a term by each term within a parenthesis. The video implicitly uses this property when 'expanding' or 'checking' the factored form of the trinomial to ensure it matches the original expression.

💡FOIL Method

The FOIL method is a mnemonic for multiplying binomials, standing for First, Outer, Inner, Last. While the video focuses on the Box method, it also uses FOIL as a means to verify the correctness of the factoring by expanding the binomials back into the original trinomial.

💡Descartes' Rule of Signs

Although not explicitly mentioned in the script, Descartes' Rule of Signs is a technique that can predict the number of positive or negative real roots a polynomial has. The video's focus on factoring trinomials is related to finding its roots, which Descartes' Rule could help estimate.

💡Polynomial

A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The video's main theme is factoring a specific type of polynomial, the trinomial, using the Box method.

Highlights

Introduction to the Box Method for factoring trinomials, an alternative to the AC method.

Explanation of how the Box Method simplifies the factor by grouping step.

Example of factoring a trinomial: 9x^2 - 21x - 8.

Step-by-step guide to writing the trinomial in descending order of powers.

Using the AC method to find values of a and C, and determining the product and sum for B.

Setting up the Box Method with a triangle to find two numbers that meet specific multiplication and addition criteria.

Placing the x^2 term and the constant term in the box as per the Box Method.

Inserting the two numbers found into the box with the appropriate x terms.

Factoring the top row of the box by finding the greatest common factor.

Determining the factors for the side of the box by identifying common factors between the terms.

Multiplying the factors to confirm the original trinomial is correctly factored.

Verification of the factored form using FOIL (First, Outer, Inner, Last) method.

Emphasizing the importance of including x terms in the box for accurate factoring.

Advantages of the Box Method over traditional factoring techniques.

Additional example to practice the Box Method with the trinomial 6x^2 + 13x - 5.

Guidance on how to handle different signs and coefficients when setting up the box.

Final check of the factored form to ensure accuracy.

Conclusion on the effectiveness of the Box Method for factoring trinomials.

Transcripts

in this video we're going to look at a

different method for factoring

trinomials and I'm going to call this

the box method this is actually a

variation of the AC method if you're

familiar with the AC method but with the

Box method it kind of takes out the

factor by grouping step which we can be

confusing for some people so let's take

a look at this example we have nine x

squared minus eight minus 21x and what

we want to do is factor that so the

first thing that you always want to do

is write your trinomial in descending

order of powers so we need to write this

as 9x squared minus 21x minus eight and

a lot of times they come in descending

power already but sometimes they don't

so like the AC method we need the values

of a and C let's let's refresh ourselves

on what the AC method is so if you have

a trinomial in this descending order ax

squared plus BX plus C a is the

coefficient on the x squared term B is

the coefficient on the X term and C is

the constant term in the AC method your

first step is to multiply a times C and

then figure out two numbers that will

multiply to be that product and add to

be the value of of B add to come up with

the B value the coefficient on the

x-term

so it's going to be the same step with

the Box method over here in our example

then our a value is nine our B value is

negative twenty-one and our C value is

negative eight we want to take a times C

we're going to set up this little

triangle thing here so a times C nine

times negative eight would be negative

72 so we have to figure out two things

that multiply together to be negative 72

and add to be negative twenty-one

so that might take a little fishing

around with negative 72 but if you work

on it for a little bit

you are going to come up with negative

24 and 3 those add to be negative 21 and

if you multiply them together you get

negative 72 so so far this is the same

as the AC method now here's where it

goes different so with the Box method

what you do is you take your first term

which is going to be your x squared term

and you always write that in the upper

left hand corner of this box so this box

is always going to get set up the same

it's a you know a rectangle cut into

four pieces in the lower right hand

corner you're always going to put your

constant term which in this case is

negative eight now after you do this

little triangle step and you come up

with your negative 24 and your three

what you do is you put those in the

other two boxes and it doesn't matter

what order you put them in you could put

them either way and you put an X term on

them okay so I have my x squared term

here and my 8 here I'm going to take my

negative 21x and I'm going to split it

up to negative 24 X + 3 X and I'm going

to put those in these boxes so I'm going

to put my 3x here and my negative 24 X

here and like I said you could switch

these and it wouldn't matter alright so

now what you do let's do this in a

different color is you look at the 9 x

squared plus the 3 X this top row I'm

saying plus because this is positive and

I ask myself what's the greatest common

factor between these two terms and they

have a 3 in common and they have an X in

common so I'm going to write that

outside here now on the top of the box

I'm simply going to write what would

multiply with 3x to give 9 x squared and

that would be 3x so if you think of this

as like the area of a rectangle 3x times

3x is 9x squared so this distance right

here I'm thinking this as 3x all right

so this is 3x and what would I have to

multiply 3x by to get 3x well that would

be a positive 1 so I put plus 1 up there

so now if you think of multiplying or

distributing 3x times 3x is 9x squared

and 3x times 1 is 3x it's kind of like

taking out a greatest common factor all

right well now we can take this 3x right

here which also corresponds to this

length and we can ask ourselves

3x times what would give us negative 24

X and that would be negative 8 and you

can also see that that's the greatest

common factor between these two terms on

the bottom row so you could also ask

yourself what's the greatest common

factor between these two terms on the

bottom row and that would be negative 8

and if I take that negative 8 out or

divide it out of negative 24x I would

get a positive 3x if I take it and

divide it out of the negative 8 I get a

positive 1 so if you see these things

all multiplied together like little

areas negative 8 times 3x is negative

24x negative 8 times positive 1 is

negative 8 and we talked about the top

row before so what you have in terms of

area if you look at the length of the

top you have 3x plus 1 and if you look

at the width of the side here or you

could think of it as the height these

parentheses are going to look a little

weird but that's okay we have 3x minus 8

and these two things multiplied together

to give this area so in other words we

know that 3 X plus 1 times 3x minus 8

will give us this area which corresponds

to this trinomial so there it is it's

factored all right so let's look at

those steps one more time before we do

another example and then we should

probably check this but it's kind of

checked right here um make sure you're

in descending order set up your AC

method just like you did before take a

times C figure out what multiplies to be

AC and adds to give B set up your box

put your x squared term in the upper

left your constant term in the lower

right your two numbers you found down

here with X's that's key they got to

have X's in these two boxes and then you

do a greatest common factor of each row

and it should correspond to these

column values as well so that they

multiply together to give the area of

each individual box what's on the top

and the side on the outside are your

factors that multiply together to give

you a trinomial so if we wanted to check

this this is our answer this is factored

if we wanted to check this we would just

foil so 3x times 3x is 9x squared 3x

times negative 8 is negative 24x one

times 3x is positive 3x one times

negative 8 is negative 8 and if you'll

notice these four terms correspond to

the four terms in the box not a

coincidence and then you could combine

like terms and when you're combining

these two like terms together you're

just basically doing what you did over

here you you pick negative 24 and 3

because they combine to be negative 21

and so you end up with the correct

trinomial all right let's do one more

example it's always good to do multiple

examples let's see and if you think you

got the hang of this you might pause the

video and try it yourself let's delete

this here let's see okay I need my

keyboard all right and we'll delete this

down here

let's try

6x squared so we're going to factor 6x

squared plus 13x minus five all right so

if you want to take a shot at this go

ahead and pause the video and then start

it when you have your answer so the

first thing we need to do is what we

could set up our box or we could do the

AC method thing I'll do the AC method

thing first so my a value is 6 my B

value is 13 my C value is negative 5 so

a times C 6 times negative 5 would be

negative 30 I need to come up with two

numbers that add to be negative 30 I'm

sorry multiply it to be negative 30 and

add to be 13 positive 13 and that is

going to be 15 and a negative 2 15 times

negative 2 is negative 30 15 plus

negative 2 is 13 so now I have all the

information I need to set up my box so

I'm going to set up a box so I would do

it here a purple one okay and we'll put

a line this way and a line this way and

in the upper left-hand corner is going

to go my x squared term in the lower

right hand corner is going to go my

constant term and you need those signs

if it's a minus 5 you got to have that

negative 5 and then I'm going to put the

15 and the negative 2 in one of these

two places here I'm going to put the

negative 2x here and the positive 15 X

here and you have to remember to put

these X's because what you're really

doing is you're breaking up this net

this 13 X term into 15 X minus 2 X okay

so now we need to do the factoring

business what goes into both 6x squared

and negative 2x well 2 goes into 2 and 6

and they both have an X so now I'll

figure out what goes on the top 2x times

what is 6x

squared that would be 3 X 2 X times what

is negative 2 X that would be a negative

1 now I can use these top values that

I've figured out to figure out this side

value although you could say what do 15

x & 5 have in common which should be 5

but you could also say that whatever I

put here times 3 X has to be 15 X well

that's going to have to be a 5 5 times 3

X is 15 X and then you can check and

make sure that it works for the last one

5 times negative 1 is negative 5 so you

have your outside values on the top you

have 3 X minus 1 on the side you have 2

X plus 5 and those are your two

binomials that multiply together to give

6 x squared plus 15 X minus 5 so that's

kind of nifty it takes out the whole

factor by grouping step which you know a

lot of people might like so let's check

this just to be sure

although the check is right here if you

can see it if I foil these I get 3x

times 2x is 6x squared right there 3x

times 5 is positive 15 X right there

negative 1 times 2x is negative 2x right

there negative 1 times 5 is negative 5

which we have there and if you combine

the two middle terms together you will

get 6x squared plus 13x minus 5 so there

it is

I hope that helps and that maybe this

box method might save you some trouble

if you're having trouble factoring these

more difficult trinomials the other way

remember this really is the best method

to use if your a value right here is not

1 if your a value is 1 if you have a 1x

squared much easier way to factor this

which hopefully you'll see in a

different video um you could use the box

method though this will work for it for

any of these type of trinomials