Factoring Tough Trinomials Using the Box Method
Summary
TLDRThis 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
- π The video introduces the 'Box Method' for factoring trinomials, which is a variation of the AC method.
- π The Box Method simplifies the process by eliminating the need for factor by grouping, which can be confusing for some.
- π The method begins by writing the trinomial in descending order of powers, ensuring the equation is set up correctly.
- π’ 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.
- π 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'.
- π The trinomial's middle term is split into two terms that correspond to the numbers in the box, ensuring each has an 'x'.
- 𧩠The greatest common factor (GCF) of each row in the box is determined, which will correspond to the factors outside the box.
- π 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.
- π The method is demonstrated through examples, showing how to factor trinomials like 9x^2 - 21x - 8 and 6x^2 + 13x - 5.
- π 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
π 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.
π 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.
π 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.
Mindmap
Keywords
π‘Factoring
π‘Trinomials
π‘Box Method
π‘AC Method
π‘Coefficient
π‘Greatest Common Factor (GCF)
π‘Distributive Property
π‘FOIL Method
π‘Descartes' Rule of Signs
π‘Polynomial
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
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