Transcript Factor_Trinomial_original

Factoring Quadratic Trinomials
…beyond the guess and test
method.
Topics
1.
2.
3.
4.
5.
6.
7.
Standard Form
When c is positive and b is positive
When c is positive and b is negative
When c is negative
When the trinomial is not factorable
When a does not equal 1
When there is a GCF
Standard Form
The standard form of any
2
ax
 bx  c
quadratic trinomial is
So, in 3x  4x  1...
2
a=3
b=-4
c=1
Now you try.
 x  7x  2
2
a = ??
b = ??
c = ??
Click here when you are ready to check your answers!

Recall
The standard form of any
2
ax
 bx  c
quadratic trinomial is
So, in 2x 2  x  5...
a=2
b = -1
c=5
Try Another!
4x  x  2
2
a = ??
b = ??
c = ??
Go on to factoring!
Factoring when a=1 and c > 0.
First list all the factors of c.
x
2
 8x  12
1
12
2
6
3
4
Find the pair that adds to ‘b’
1
12
2
6
3
4
These numbers are used in the
factored expression.
x
 2x  6
Now you try.
1.
2.
3.
x
2
 8x  15
x
2
 10x  21
x
2
 9x  20
Click here when you are ready to check your answers!
Recall
x
2
 10x  24
We need to list the
factors
 of c.
So we get:
x
 4x  6
1
24
2
12
3
8
4
6
Try some others!
x  6x  9
2
1.
(x+3)(x+3)
2.
(x+2)(x+3)
2.
x 2  8x  7
x  7x  6
2
(x+1)(x+6)
(x+2)(x+3)
2.
x 2  8x  7
Go on to factoring where b is negative!
Factoring when c >0 and b < 0.
Since a negative number times a
negative number produces a positive
answer, we can use the same method.
Just remember to use negatives
in the expression!
Let’s look at
x  13x  12
2
First list the factors of 12
1
12
We need 
a sum of -13
2
6
3
4
Make sure both values are negative!
x
 12x  1

Now you try.
1.
2.
3.
x
2
 5x  4
x
2
 9x  14
x
2
 13x  42
Click here when you are ready to check your answers!
Recall
x  6x  8
2
In this case, one factor
should be positive and the
other negative.
1
8
2
4
We need a sum of -6
x
 2x  4
Try some others!
1.
x  7x  12
2
(x-3)(x-4)
2.
(x-3)(x+4)
2.
x 2  8x  7
x  4x  4
2
(x-2)(x-2)
(x-1)(x-4)
2.
x 2  8x  7
Go on to factoring where c is negative!
Factoring when c < 0.
We still look for the factors of c. However, in
this case, one factor should be positive and the
other negative.
Remember that the only way we can
multiply two numbers and come up
with a negative answer, is if one is
number is positive and the other is
negative!
Let’s look at
x  x  12
2
In this case, one factor
should be positive and the 1
other negative.
2

We need a sum of -1
x
 3x  4
3
12
6
4

Now you try.
1.
2.
3.
4.
x
2
 3x  4
x
2
 x  20
x
2
 4 x  21
x
2
 10x  56
Click here when you are ready to check your answers!
x  3x  18
Recall
2
In this case, one factor
should be positive and the 1
other negative.

2
We need a sum of 3
x
3
18
12
6
 3x  6
Try some others!
1. x  2x 15
2
(x-3)(x+5)
2.
(x+3)(x-5)
2.
x 2  8x  7
x  x  30
2
(x-5)(x+6)
(x-6)(x-5)
2.
x 2  8x  7
Go on to trinomials that are not factorable
Prime Trinomials
Sometimes you will find a quadratic
trinomial that is not factorable.
You will know this when you
cannot get b from the list of
factors.
When you encounter this
write not factorable or
prime.
Here is an example…
x  3x  18
2
1
18
2
9
3
6
Since none of the pairs adds to 3,
this trinomial is prime.
Now you try.
x
2
 6x  4
factorable
prime
x
2
 10x  39
factorable
prime
x
2
 5x  7
factorable
prime
Go on to factoring when a≠1
When a ≠ 1.
Instead of finding the factors of c:
Multiply a times c.
Then find the factors of this product.
1
70
2
7x  19x  10
a  c  70
2
35
5
14
7
10
We still determine
the factors that add
to b.
So now we have
x
 5x  14
1
70
2
35
5
14
7
10
But we’re not finished yet….
Since we multiplied in the beginning,
we need to divide in the end.
Divide each constant by a.
 5  14 
x  x  
 7 
7 
Simplify, if possible.
 5 
x  x  2
 7 

Clear the fraction in each
binomial factor

7x  5x  2
Recall
2x  3x  9
2
Multiply a times c.
List factors.
Write 2 binomials with
the factors that add to b
Divide each constant by a.
Simplify, if possible.
Clear the fractions in each factor
2  9  18
1
18
2
3
9
6
x  6x  3
6 
3

 x   x  
2 
2

3

x  3 x  
2

x  32x  3
Try some others!
Now you try.
1.
4x 4x  3
2.
3x  5x  12
3.
6x  23x  7
2
2
2
Click here when you are ready to check your answers!
1.
2x  9x  5
2
(2x-1)(x+5)
2.
(2x+5)(x+1)
2.
x 2  8x  7
4x  6x  5
2
(2x-5)(2x+1)
(4x+5)(x-1)
2.
x 2  8x  7
Go on to trinomials that have a GCF
Sometimes there is a GCF.
If so, factor it
out first.
2  15  30
1
2
3
5

30
15
10
6
Ex) 4x  2x  30
2
22x 2  x  15
2x  6x  5

6 
5 
2x  x  

2 
2 

5 
2x  3x  

2 
2x  35x  2
Now you try.
1. 4 x 2  16x  12
2. 6x
2
 10x  6

Click here when you are ready to check your answers!
Recall
45x  35x  10
2
First factor out the GCF.
Then factor the
remaining trinomial.
9 times 2 = 18
1
2
3
18
9
6


5 9x  7x  2
2
5x  2x  9
2 
9

5 x   x  
9 
9

59x  2x  1
59x  2x  1
Try some others!
1.
6x  30x  36
2
6(x-1)(x+6)

2.
2.
x 2  8x  7
4x  14x  10
2
2(2x+1)(x+5)

(6x+6)(x-6)
2(2x+5)(x+1)
2.
x 2  8x  7
Did you get these answers?
a  1
b 7
c  2
Yes
No
Did you get these answers?
1.
x  3x  5
2.
x  3x  7
x  5x  4
3.
Yes
No
Did you get these answers?
1.
x  1x  4
2.
x  2x  7
x  6x  7
3.
Yes
No
Did you get these answers?
1.
2.
3.
4.
x  1x  4 
x  4 x  5
x  3x  7
x  4 x  14 
Yes
No
Did you get these answers?
1.
2.
3.
2x  12x  3
x  33x  4 
3x  12x  7
Yes
No
Did you get these answers?
1. 4x  1x  3
2. prime
Yes
No
Good Job!
You have completed
Standard Form!
Good Job!
You have completed factoring
“When c is positive and b is
positive”!
Good Job!
You have completed
factoring “When c is positive and
b is negative”!
Good Job!
Good Job!
You have completed factoring
“When a does not equal 1”!
Good Job!
You have completed factoring
“When c is negative”!
Good Job!
Good Job!
Good Job!
Good Job!
Good Job!
Good Job!
You have completed factoring
“When there is a GCF”!
Review and Try Again!
Review and Try Again!
Review and Try Again!
Review and Try Again!
Review and Try Again!
Try Again!
Try Again!
Try Again!
Try Again!
Try Again!
Review and Try Again!
Try Again!
Review and Try Again!