Factoring Quadratic Trinomials …beyond the guess and test method.
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Transcript of Factoring Quadratic Trinomials …beyond the guess and test method.
Factoring Quadratic Trinomials
…beyond the guess and test method.
Topics
1. Standard Form2. When c is positive and b is positive3. When c is positive and b is negative4. When c is negative5. When the trinomial is not factorable6. When a does not equal 17. When there is a GCF
The standard form of any quadratic trinomial is
Standard Form
cbxax ++2
€
So, in 3x 2 − 4x + 1... a=3
b=-4
c=1
Now you try.
272 −+− xx
a = ??
b = ??c = ??
Click here when you are ready to check your answers!
The standard form of any quadratic trinomial is
Recall
cbxax ++2
€
So, in 2x 2 − x + 5... a = 2
b = -1
c = 5
Try Another!
24 2 −+ xxa = ??
b = ??c = ??
Go on to factoring!
Factoring when a=1 and c > 0.
First list all the factors of c.
€
x 2 + 8x + 121 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.
€
x 2 + 9x + 20€
x 2 + 8x + 15
€
x 2 + 10x + 21
Click here when you are ready to check your answers!
1.
2.
3.
Recall
€
x 2 + 10x + 24
1 24
2 12
3 8
4 6So we get:
€
x + 4( ) x + 6( )
Try some others!
We need to list the factors of c.
96.1 2 ++ xx
(x+3)(x+3) 78.2 2 ++ xx(x+2)(x+3)
67.2 2 ++ xx
(x+1)(x+6) 78.2 2 ++ xx(x+2)(x+3)
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 2 − 13x + 12
1 12
2 6
3 4
€
x − 12( ) x − 1( )
We need a sum of -13
Make sure both values are negative!
First list the factors of 12
Now you try.
€
x 2 − 5x + 4
x 2 − 9x + 14
x 2 − 13x + 42
Click here when you are ready to check your answers!
1.
2.
3.
Recall 862 +− xx
1 8
2 4
In this case, one factor should be positive and the other negative.
€
x − 2( ) x − 4( )
We need a sum of -6
Try some others!
127.1 2 +− xx
(x-3)(x-4) 78.2 2 ++ xx(x-3)(x+4)
44.2 2 +− xx
(x-2)(x-2) 78.2 2 ++ xx(x-1)(x-4)
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 2 − x − 12
1 12
2 6
3 4
In this case, one factor should be positive and the other negative.
€
x + 3( ) x − 4( )
We need a sum of -1
Now you try.
€
x 2 + 3x − 4
x 2 + x − 20
x 2 − 4x − 21
x 2 − 10x − 56
Click here when you are ready to check your answers!
1.
2.
3.
4.
Recall
€
x 2 + 3x − 18
1 18
2 12
3 6
In this case, one factor should be positive and the other negative.
€
x − 3( ) x + 6( )We need a sum of 3
Try some others!
€
1. x 2 − 2x −15
(x-3)(x+5) 78.2 2 ++ xx(x+3)(x-5)
30.2 2 −+ xx
(x-5)(x+6) 78.2 2 ++ xx(x-6)(x-5)
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 2 + 3x + 18 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
€
x 2 − 10x − 39
€
x 2 + 5x − 7
factorable prime
factorable prime
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.
€
7x 2 − 19x + 10
a × c = 70
1 70
2 35
5 14
7 10
1 70
2 35
5 14
7 10
We still determine the factors that add to b.
So now we have
But we’re not finished yet….
€
x − 5( ) x − 14( )
Since we multiplied in the beginning, we need to divide in the end.
€
x −5
7
⎛
⎝ ⎜
⎞
⎠ ⎟ x −
14
7
⎛
⎝ ⎜
⎞
⎠ ⎟
€
x −5
7
⎛
⎝ ⎜
⎞
⎠ ⎟ x − 2( )
Divide each constant by a.Simplify, if possible.
€
7x − 5( ) x − 2( )Clear the fraction in each binomial factor
Recall
Divide each constant by a.
Simplify, if possible.
Clear the fractions in each factor
Multiply a times c.
List factors.
Write 2 binomials with the factors that add to b
932 2 −− xx
1892 =×
63
92
181
( )( )36 +− xx
⎟⎠
⎞⎜⎝
⎛ +⎟⎠
⎞⎜⎝
⎛ −23
26
xx
( ) ⎟⎠
⎞⎜⎝
⎛ +−23
3 xx
( )( )323 +− xx
Try some others!
Now you try.
7236.3
1253.2
344.1
2
2
2
+−
−−
−+
xx
xx
xx
Click here when you are ready to check your answers!
592.1 2 −+ xx
(2x-1)(x+5) 78.2 2 ++ xx(2x+5)(x+1)
564.2 2 −− xx
(2x-5)(2x+1) 78.2 2 ++ xx(4x+5)(x-1)
Go on to trinomials that have a GCF
Sometimes there is a GCF.
If so, factor it out first.
3024x Ex) 2 −− x
€
2 2x 2 − x − 15( )
€
2 × 15 = 301 302 153 105 6
€
2 x − 6( ) x + 5( )
€
2 x −6
2
⎛
⎝ ⎜
⎞
⎠ ⎟ x −
5
2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
2 x − 3( ) x −5
2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
2 x − 3( ) 5x + 2( )
Now you try.
€
4x 2 + 16x + 12
6x 2 + 10x + 6
Click here when you are ready to check your answers!
1.
2.
Recall
First factor out the GCF.
103545x 2 −− x
( )2795 2 −− xx
118
2 93 6
( )( )925 −+ xx
⎟⎠
⎞⎜⎝
⎛ −⎟⎠
⎞⎜⎝
⎛ +99
92
5 xx
( )( )1295 −+ xx
( )( )1295 −+ xx
Then factor the remaining trinomial.
9 times 2 = 18
Try some others!
€
6x 2 + 30x − 36
6(x-1)(x+6) 78.2 2 ++ xx(6x+6)(x-6)
€
4x 2 + 14x + 10
2(2x+1)(x+5) 78.2 2 ++ xx2(2x+5)(x+1)
1.
2.
Did you get these answers?
2
7
1
−==−=
cba
Yes No
Did you get these answers?
( )( )45.3 ++ xx
( )( )53.1 ++ xx
( )( )73.2 ++ xx
Yes No
Did you get these answers?
( )( )76.3 −− xx
( )( )41.1 −− xx
( )( )72.2 −− xx
Yes No
Did you get these answers?
( )( )( )( )( )( )( )( )144.4
73.3
54.2
41.1
−+
−+
+−
+−
xx
xx
xx
xx
Yes No
Did you get these answers?
( )( )( )( )( )( )7213.3
433.2
3212.1
−−
+−
+−
xx
xx
xx
Yes No
Did you get these answers?
( )( )prime.2
314.1 ++ xx
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!