Transcript Chapter 6 Supplement on Factoring

Factoring
Greatest Common Factor
The largest common factor in all the
terms of the polynomial
Example
• pg. 342, # 9
21x  35x  14x
5
4
GCF is 7x
Factors as
3
3
7x (21x  5x  2)
3
2
Factoring by Grouping
See page 341
Example
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Factor: (mx +3qx+my+3qy)
We will take the terms in pairs
The GCF of mx and 3qx is x
The GCF of my and 3qy is y
Example 39 on page 342
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Factor: mx +3qx+my+3qy
Step 1: Use GCF in pairs: x(m+3q) + y( m+3q)
Step 2: Notice or look for any common factors
x(m+3q) + y( m+3q)
Step 3: Factoring out the common factors yields
The factored form is: (m+3q)(x+y)
Special
Factorizations
Strategy for
Factoring Polynomials
Solving Equations by Factoring
Solve the equation:
See page 366 # 18
(3x  1)(x  3)  2  3(x  5)
The Plan
• Distribute (FOIL) the left hand side (LHS)
• Distribute the right hand side (RHS)
• Gather all terms on the RHS, setting the LHS
equal to 0
• Factor ( if possible) the LHS
• Apply the Zero-Factor Property to Solve (see page
360)
• Check our solution in the ORIGINAL Problem
Distribute on RHS and LHS
(3x  1)(x  3)  2  3(x  5)
 3x  9x  x  3  2  3x  15
2
 3x  8x  3  3x  17
2
Gather all terms on the LHS,
setting the RHS equal to 0
3x  8x  3  3x  17
2
 3x  8x  3  3x  17  0
2
 3x  11x  20  0
2
Factor ( if possible) the LHS
• Remember that we must use trial
and error to find the factors until we
exhaust all possibilities
3x  11x  20  0
 (3x  4)(x  5)  0
2
Apply the Zero-Factor Property
to Solve
(3x  4)(x  5)  0
3x  4  0
3x  4
x 
4
3
OR
x 5 0
x 5
x=5
The Check
•
For
x 
4
3
(3x  1)(x  3)  2  3(x  5)
4
4
4
(3  [  ]  1)( 
 3)  2  3( 
 5)
3
3
3
4
4
4
( 3  [
]  1)( 
 3)  2  3( 
 5)
3
3
3
13
11
( 4  1)( 
)  2  3(
)
3
3
13
3( 
)  2  11
3
13 = 13
The Check continued
For
x 5
(3x  1)(x  3)  2  3(x  5)




(3  [5]  1)(5  3)  2  3(5  5)
(15  1)(2)  2  3(10)
(16)(2) = 2+30
32
= 32
Solving a Quadratic Equation by
Factoring Overview
• Step 1: Use algebraic properties to ensure that one
side is equal to 0
• Step 2: Factor this polynomial
• Step 3: Use the zero-factor property by setting
each of the factors equal to 0
• Step 4: Solve each of the equations from step 3
independently
• Step 5: Check solution in the original problem
Good Luck!
• Only three chapters away from completing
the course.