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April 7, 2009
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HW answers: p.607
13. B
25. (x – 15)(x – 30)
16. (t – 3)(t – 7)
29. (x -2)(x – 7)
19. (y – 6)(y + 3)
22. (4 + n)(8 + n)
roots = 2 and 7
34. (x + 7)(x – 8)
roots = -7 and 8
41. (x – 2)(x – 9)
roots = 2 and 9
10.6 Factoring ax2 + bx + c (where a ≠
1)
In this section, all of the trinomials will have
either a positive or negative leading
coefficient.
The textbook has a method that works
based on guess and check – look at page
611 right now.
Now look at page 612.
It sometimes makes you do a lot of work!
My way to solve ax2 + bx + c is called
SLIDE and DIVIDE
To factor the problem in Example #3 (p.612),
6x2 – 19x + 15
First SLIDE - slide the leading coefficient
over to the c term and multiply.
6X2 – 19x + (15 * 6)
Your newly created trinomial looks like:
x2 – 19x + 90
Now you can factor the trinomial like yesterday
(find two numbers that multiply to equal 90m,
but also add up to -19).
Factors of 90:
1, 90
2, 45
3, 30
5,18
6, 15
9, 10
-1,-90
-2,-45
-3,-30
-5,-18
-6,-15
-9,-10
Which number add up to -19?
You hopefully picked -9 and -10 so
(x – 9) (x – 10)
Second, DIVIDE – place the original leading
coefficient (6) under the numbers you chose
(like the 6 is dividing):
(x – 9) (x – 10)
6
6
Now, simplify each fraction:
(x – 3) (x – 5)
2
3
You’re almost done!
Since neither fraction simplified to a whole
number, MOVE the denominator in front of
each x :
(x – 3) (x – 5)
2
3
(2x – 3) (3x – 5)
This is your factored form of 6x2 – 19x + 15.
CHECK
Use FOIL or Dist.Prop. to see if you get the original
trinomial back.
Here’s another example of Slide and Divide:
Factor 2x2 + 7x + 6.
SLIDE
1.Slide the lead. coeff. (2) over to the c (6)
and multiply.
X2 + 7x + (2 * 6)
X2 + 7x + 12
2. Now factor the trinomial (find 2 numbers
that mult. to 12 and add up to 7)
You now have (x + 3) (x + 4)
[or the reverse]
DIVIDE
1. Place the original lead. coeff. under each
number in the ( ).
(x + 3) (x + 4)
2
2
2. Now simplify each fraction.
(x + 3) (x + 2)
2
Because the 3/2 didn’t simplify to a whole
number, MOVE the 2 up in front of the x:
(2x + 3) (x + 2)
CHECK by FOIL or Dist. Prop.
Now find the roots – set each ( ) equal to zero
(2x + 3) = 0
(x + 2) = 0
2x + 3 = 0
x+2=0
x = -3/2
x = -2
The roots to the parabola 2x2 + 7x +6 are
-3/2 and -2.
Confused again? Get someone to help
you who understands how to Slide and
Divide.
Practice:
p.614, #5-8 matching
Homework:
p.614, #17, 21, 23, 24, 28, 30, 31