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
F
IRST terms in the brackets.
(
x
2 )(
x
4 )
x
2
Step 2: Multiply the
O
UTSIDE terms.
(
x
2 )(
x
4 )
x
2 4
x
Step 3: Multiply the
I
NSIDE terms.
(
x
2 )(
x
4 )
x
2 4
x
2
x
Step 4: Multiply the
L
AST terms.
(
x
2 )(
x
4 )
x
2 4
x
2
x
8
Step 5: Collect like terms.
x
2 4
x
2
x
8
x
2 6
x
8
( 3
x
2 )(
x
4 ) ( 3
x
2 )(
x
4 ) 3
x
2 12
x
2
x
8 3
x
2 14
x
8
( 3)(
x
6) (6
x
7)(2
x
2)
Math 10-C George McDougall High School
Factoring Simple Trinomials
x
2 + 10 x + 16 = (x + 2)(x + 8)
Check by FOILing
= x 2 + 8x + 2x + 16 = x 2 + 10x + 16
x
2 + 9x + 20 = (x + 5)(x + 4)
x
2 + 5x + 4 = (x + 4)(x + 1)
x
2 + 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) = x 2 + 3x + 4x + 12 = x 2 + 7x + 12
The
middle term
of a simple trinomial is the
SUM
last two terms of the binomials.
of the The
last term
of a simple trinomial is the
PRODUCT
the last two terms of the binomials.
of Therefore this type of factoring is referred to as
SUM-PRODUCT
!
To factor trinomials, you ask yourself…
x12 +7
1,12 13 2,6 8 3,4 7
x
2
+ 7x + 12
(x + 3)(x + 4)
Factor:
x
2
– 8x +12
( x – 2)( x – 6)
x 12
1, 12 -1, -12 2, 6 -2, -6
– 8
13 -13 8 -8
Factor:
m
2
– 5m -14
(m + 2) (m – 7)
x (-14)
-1, 14 1, -14 -2, 7 2, -7
-5
13 -13 5 -5
Factor:
x
2
- 11x + 24 x
2
+ 13x + 36 x
2
- 14x + 33
Factor: x 2 + 12x + 32 x 2 - 20x + 75 x 2 + 4x – 45 x 2 + 17x + 72 x 2 - 7x – 8
Factor:
- 5t – 3t
2
+ 15 + 4t
2
– 3 - 3t
STEP 1: Combine Like terms t 2 – 8t +12 ( x – 2 )( x – 6) x 12 1, 12 -1, -12 2, 6 -2, -6 - 8 13 -13 8 -8
Factor:
7q
2
– 14q - 21
7 ( q 2 –2q –3) 7 ( q – 3)( q + 1) STEP 1: Pull out the GCF -3 -1, 3 -3, 1 -2 2 -2
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(
x
2 7
x
10)
common factor?
2(
x
5)(
x
2) + = 7 x = 10 5, 2 ex.
3
x
2 3
x
60 3(
x
2 20)
common factor?
3(
x
4)(
x
5) + = 1 x = -20 -4, 5
How could we factor this using algebra tiles? 1.
2.
Create a rectangle using the exact number of tiles in the given expression.
Remember that a trinomial represents area – two binomials multiplied together.
3.
4.
What is the width and length of the rectangle?
These are the FACTORS of the original rectangle.
x + 3 Does that make sense?
x + 2 (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?
x
4. These are the FACTORS of the original rectangle.
2 3
x
2
x
2 4
x
4