Transcript Factoring Trinomials of the form x2 + bx + c

Math 10-C
George McDougall High School
NOTE: Props to Ms. Somerville for
allowing me to adapt your notes!
 Step
1: Multiply the
FIRST terms in the brackets.
( x  2)(x  4)
x
2
Step
2: Multiply the
OUTSIDE terms.
( x  2)(x  4)
 x  4x
2
Step
3: Multiply the INSIDE terms.
( x  2)(x  4)
 x  4x  2x
2
Step
4: Multiply the LAST terms.
( x  2)(x  4)
 x  4x  2x  8
2
Step
5: Collect like terms.
 x  4x  2x  8
2
 x  6x  8
2
(3x  2)(x  4)
(3x  2)(x  4)
 3x  12 x  2 x  8
2
 3x  14x  8
2
( x  3)( x  6)
(6 x  7)(2 x  2)
Math 10-C
George McDougall High School
Factoring Simple Trinomials
x2 + 10 x + 16 = (x + 2)(x + 8)
Check by FOILing
x2 + 9x + 20
= (x + 5)(x + 4)
x2 + 5x + 4
= (x + 4)(x + 1)
= x2 + 8x + 2x + 16
= x2 + 10x + 16
x2 + 11x + 24
= (x + 8)(x + 3)
What relationship is
there between
product form and
factored form?
Factoring Simple Trinomials
Many trinomials can be written as
the product of 2 binomials.
Recall: (x + 4)(x + 3) = x2 + 3x + 4x + 12
= x2 + 7x + 12
The middle term of a simple trinomial is the SUM of the
last two terms of the binomials.
The last term of a simple trinomial is the PRODUCT of
the last two terms of the binomials.
Therefore this type of factoring is referred to as
SUM-PRODUCT!
To factor trinomials, you ask yourself…
x12
1,12
2,6
3,4
+7
13
8
7
2
x
+ 7x + 12
(x + 3)(x + 4)
Factor:
x2 – 8x +12
( x – 2)( x – 6)
x 12
–8
1, 12 13
-1, -12 -13
2, 6
8
-2, -6 -8
Factor:
m2 – 5m -14
(m + 2) (m – 7)
x (-14)
-5
-1, 14
1, -14
-2, 7
2, -7
13
-13
5
-5
Factor:
x2 - 11x + 24
2
x
+ 13x + 36
x2 - 14x + 33
Factor:
2
x
+ 12x + 32
x2 - 20x + 75
x2 + 4x – 45
x2 + 17x + 72
x2 - 7x – 8
Factor:
- 5t – 3t2 + 15 + 4t2 – 3 - 3t
STEP 1:
Combine
Like terms
x 12 - 8
1, 12 13
-1, -12 -13
2, 6 8
-2, -6 -8
t2 – 8t +12
( x – 2)( x – 6)
Factor:
7q2 – 14q - 21
7 ( q2 –2q –3)
7 ( q – 3)( q + 1)
-3
-2
-1, 3 2
-3, 1 -2
STEP 1:
Pull out the
GCF
To Summarize:
1. Always check to see if you can simplify first!
2. Then check to see if you can pull out a common
factor.
3. Write 2 sets of brackets with x in the first
position.
4. Find 2 numbers whose sum is the middle
coefficient, and whose product is the last term.
5. Check by foiling the factors.
ex.
2 x 2  14 x  20
 2( x2  7 x  10)
common factor?  2( x  5)( x  2)
2
3
x
 3x  60
ex.
 3( x  x  20)
2
common factor?  3( x  4)( x  5)
+=7
x = 10
5, 2
+= 1
x = -20
-4, 5
How could we factor this using algebra tiles?
1. Create a rectangle using the exact number of tiles in the
given expression.
2. Remember that a trinomial represents area – two binomials
multiplied together.
3. What is the width and length of the rectangle?
4. These are the FACTORS of the original rectangle.
x+3
x+2
Does that
make
sense?
(x+3)(x+2)
1. Create a rectangle using the exact number of
tiles in the given expression.
2. Remember that a trinomial represents area –
two binomials multiplied together.
3. What is the width and length of the
rectangle?
4. These are the FACTORS of the original
rectangle.
x  3x  2
2
x  4x  4
2