Algebra (Periods 6 & 7)

 Factoring Pattern for ax² + bx+ c  Section 5-9 

When a > 1
We used a different method than what is taught in the book.

I first showed you what I call the "Matrix" method

2x² + 7x - 9

Consider the last sign... in this case the negative.
What does that tell us?
"That the signs in the two sets of (  )(  ) are different."
What does the first sign tell us?  in this case we have a positive.
That the positive " wins."

Now consider all the factors of 2
That's easy  just 2 and 1
Set them in a column
2
1

Now consider the factors of 9
Hmm... that's  1 and 9 as well as 3 and 3
Now you need to set up a matrix
You can try out all the different combinations
2 3
1 3
or
2  1
1  9

or
2 9
1 1

What you do at this point is multiply diagonally
that is with the first matrix
2 3
1 3
You would multiply the upper left number (2) with the lower right number (3) = 6
You would then take the upper right number (3) and multiply it by the lower left (1) = 3
Ask yourself, is there anyway to get a difference of 7 (the middle term in your problem above)?
NO-- so that matrix is not correct.



2  1
1  9
Try the same with this
You would multiply the upper left number (2) with the lower right number (9) = 18
You would then take the upper right number (1) and multiply it by the lower left (1) = 1
Ask yourself, is there anyway to get a difference of 7 (the middle term in your problem above)?
NO-- so that matrix is not correct.

But with the last matrix
2 9
1 1
Try it
You would multiply the upper left number (2) with the lower right number (1) = 2
You would then take the upper right number (9) and multiply it by the lower left (1) = 9
Ask yourself, is there anyway to get a difference of 7 (the middle term in your problem above)?
Yes-- so that matrix is correct but which product needs to be + so we end up with +7? The Lower left product. Travel up the arrow and place the + in front of the number on the upper right! Then place the opposite sign on the number below it. AS this shows:
2 +9
1  -1

Now just read across..  and return the variable
(2x +9)(x - 1)
You should ALWAYS FOIL, Double DP, BOX and make sure you have factored correctly!

Tomorrow I will show you   " X box"

2x² + 7x - 9
Multiply the 2 and the 9
put eighteen in the box
Your controllers are
2x² and -9
THen using a T chart find the factors of 19 such that the difference is 7x
we found that +9x and -2x worked

so
2x² +9x -2x -9
Then separate them in groups of 2
such that


(2x² +9x) + (-2x -9)

Then realize you can factor a - from the second pair

(2x² +9x) - (2x + 9)
Then wht is the GCF in each of the hugs( )
x(2x +9) -1(2x +9)
look they both have 2x + 9
:)
(2x +9)(x-1)
But what if you said -2x + 9x instead to make the +7x in the middle
Look what happens
(2x² -2x) + (9x -9)
now, factor te GCF of each
2x(x -1) + 9(x -1)
now they both have x -1
(x-1)(2x +9)
SAME RESULTS!!

14x² -17x +5
remember the second sign tells us that the numbers are the same and the first sign tells us that they are BOTH negative

create your X BOX with the product of 14 and 5 in it
70

Place your controllers on either side

14x² and + 5

Now do your T Chart for 70
You will need two numbers whose product is 70 and whose sum is 17
that's 7 and 10

14x² -7x -10x + 5

Now group in pairs

(14x² -7x) + (-10x + 5)
which becomes

(14x² -7x) - (10x - 5)

FACTOR each
7x(2x -1) - 5(2x-1)
(2x-1)(7x-5)

10 + 11x - 6x ²

sometimes its better to arrange by decreasing degree so this becomes

- 6x ² +11x + 10

now factor out the -1 from each terms


- (6x ² - 11x - 10)

Se up your X BOX with the product of your two controllers :)
60 We discover that +4x and -15x are the two factors

-1(6x ² +4x - 15x - 10)

-1[(6x ² +4x) + (- 15x - 10)]
-1[6x ² +4x) - (15x +10)
-1[2x(3x +2) -5(3x+2)]
-(3x+2)(2x-5)


If you had worked it out as
10 + 11x -6x² you would have ended up factoring
(5 -2x)(2 + 3x)
and we all know that
5 -2x = -(2x-5) Right ?


Next, we looked at the book and the example of
5a² -ab - 22b²
We discussed the books instructions to test the possibilities and decided that the X BOX method was much better.... I need to check out hotmath.com... did you????

5a² -ab - 22b² Using X BOX method we have 110 in the box and the controllers are
5a² and - 22b²
What two factors will multiply to 110 but have the difference -1?
Why 10 and 11

5a² +10ab -11ab - 22b²

separate and we get
(5a² +10ab) + (-11ab - 22b²)
( 5a² +10ab) - (11ab + 22b²)

5a(a + 2b) -11b(a + 2b)
(a + 2b)(5a - 11b)