2.3 Part 1 Factoring 10/29/2012. What is Factoring? It is finding two or more numbers or algebraic...
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Transcript of 2.3 Part 1 Factoring 10/29/2012. What is Factoring? It is finding two or more numbers or algebraic...
2.3 Part 1Factoring
10/29/2012
What is Factoring?
It is finding two or more numbers or algebraic expressions, that when multiplied together produce a given product.Ex. Factor 6: 2 • 3 Factor 2x2 + 4: 2 • (x2 +2) Factor x2 +5x + 6: (x+2)•(x+3)
Type 1 Problems
Factoring Quadratic equations in Standard form
y = ax2 + bx +cwhen a = 1
and when a > 1
The Big “X” method
c
b
Think of 2 numbers that Multiply to c and Add to b
#1 #2
add
multiply
Answer: (x ± #1) (x ± #2)
Factor: x2 + bx + cNote: a = 1
15
8
Think of 2 numbers that Multiply to 15 and Add to 8
3 x 5 = 153 + 5 = 8
3 5
Answer: (x + 3) (x + 5)
Factor: x2 + 8x + 15
c
b
#1 #2
add
multiply
8
-6
Think of 2 numbers that Multiply to 8 and Add to -6
-4 x -2 = 8-4 + -2 = -6-4 -2
Answer: (x - 4) (x - 2)To check: Foil (x – 4)(x – 2) and see if you get x2-6x+8
Factor: x2 - 6x + 8
c
b
#1 #2
add
multiply
-9
8
Think of 2 numbers that Multiply to -9 and Add to 8
9 x -1 = -99 + -1 = 8
-9 8
Answer: (x - 9) (x + 8)
Factor: x2 + 8x - 9
c
b
#1 #2
add
multiply
The Big “X” method
a•c
b
Think of 2 numbers that Multiply to a•c and Add to b
#1 #2
add
multiply
Answer: Write the simplified answers in the 2 ( ) as binomials. Top # is coefficient of x and bottom # is the 2nd term
Factor: ax2 + bx + cNote: a > 1
a aSimplify like a fraction if needed
Simplify like a fraction if needed
3•2 = 6
7
Think of 2 numbers that Multiply to 6 and Add to 7
6 x 1 = 66 + 1 = 76 1
Answer: (x + 2) (3x + 1)
Factor: 3x2 + 7x + 2
a•c
b
#1 #2
add
multiply
3 3Simplify like a fraction . ÷ by 3
2
1
a a
4(-9) = -36
-16
Think of 2 numbers that Multiply to -36 and Add to -16
-18 x 2 = -36 -18 + 2 = -16
-18 2
Answer: (2x - 9) (2x + 1)
Factor: 4x2 - 16x - 9
a•c
b
#1 #2
add
multiply
4 4Simplify like a fraction . ÷ by 2
-9
2
a a1
2 Simplify like a fraction . ÷ by 2
Type 2 ProblemsFactoring Quadratic equations
written as Difference of 2 Squares.
Difference of Two Squares Pattern
(a + b) (a – b) = a2 – b2
In reverse, a2 – b2 gives you (a + b) (a – b)Examples: 1. x2 – 4 = x2 – 22 = (x + 2) (x – 2)
2. x2 – 144 =(x + 12) (x – 12)3. 4x2 – 25 = (2x + 5) (2x – 5)
If you can’t remember that, you can still use the big X method.
Factor: x2 – 4
-4
0
Think of 2 numbers that Multiply to -4 and Add to 0
2 x -2 = -42 + -2 = 0
2 -2
Answer: (x + 2) (x - 2)
Ex. x2 + 0x – 4
Ex. x2 – 144
-144
0
Think of 2 numbers that Multiply to -144 and Add to 0
12 x -12 = -14412 + -12 = 0
12 -12
Answer: (x + 12) (x - 12)
x2 + 0x – 144
4(-25) = -100
0
Think of 2 numbers that Multiply to -100 and Add to 0
-10 x 10 = -100 -10 + 10 = 0
-10 10
Answer: (2x - 5) (2x + 5)
Factor: 4x2 - 25
4 4Simplify like a fraction . ÷ by 2
-5
2
5
2 Simplify like a fraction . ÷ by 2
4x2 + 0x - 25
Type 3 ProblemsFactoring Quadratic equations by taking out the Greatest Common
Factor
Factor y = x2 – 6x
1. Find the GCF.GCF = x2. Factor the GCF out. Think reverse “distributive prop.”y = x (x – 6)
Factor y = -8x2 + 18 1. Find the GCF.GCF = -2Why -2 and not 2 you ask? Wait for the next step.
2. Factor the GCF out. y = -2 (x2 - 9)Answer: So we can have the difference of 2 squares pattern
y = 2 (-x2 + 9) Not Difference of 2 Squares3. Factor what’s in the ( ) since it follows the difference of 2 square pattern. y = -2(x – 3)(x + 3)