Algebra Honors (Period 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 showed 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)



Factoring by Grouping 5-10


5(a -3) - 2a (3 -a)

a-3 and 3-a are OPPOSITES
so we could write 3-a as -(-3 +a) or -(a -3)
sp we have
5(a-3) -2a [-(a-3)]
which is really
5(a-3) + 2a(a-3)
wait... look... OMG they both have a-3
so
(a-3)(5 + 2a)

What about
2ab-6ac + 3b -9c

What can you combine...
some saw the following:

(2ab -6ac) + 3b -9c)
then
2a(b-3c) + 3( b-3c)
(b -3c)(2a + 3)

BUT others look at 2ab-6ac + 3b -9c and saw
2ab +3b -6ac -9c
which lead them to
(2ab + 3b) + (-6ac -9c)
b(2a +3) -3c(2a +3)
(2a +3)(b-3c)
wait that's the same!!
Hooray

What about 4p² -4q² +4qr -r²
First look carefully and you will see

4p² -4q² +4qr -r²
That's a trinomial square OMG

so isn't that
4p² - ( 2q -r)²

BUT WAIT look at

4p² - ( 2q -r)² That's the
Difference of Two Squares
Which becomes
(2p + 2q -r)(2p -2q +r)