Download Review of Factoring - Winston Knoll Collegiate

Review of Factoring (With Solutions)
First we must review factoring skills to have success in this section.
Type1: Factoring by Removing the Greatest Common Factor (GCF)
Remember the Distributive Property
2(x + 1) = 2x + 2
x(x + 4) = x2 + 4x
Factoring GCF’s is the opposite of the Distributive Property
o 1st: look for the biggest number that each term can be divided by
o 2nd: look for the variables that are common to each term
Example1
Factor: 12x + 24
12(x + 2) Note: both 12 and 24 are divisible by 12
12 is the GCF of this question
Example2
Factor: 2x + 3
We cannot divide by anything other than 1, so
we call this expression “prime”
Example3
Factor: -4x + 6
-2(2x – 3) When the 1st term (the leading coefficient)
is negative, factor out the negative.
Note: -4 and 6 are divisible by -2, so
-2 is the GCF.
REMEMBER: Factoring is the opposite of the “Distributive property”.
Example4
Factor: x2 + 2x
x(x + 2)
x is common to both terms, so x is the GCF
Example5
Factor: 2x2 – 10x
2x(x – 5)
2x is common to both terms, so 2x is the GCF
Practice
Factor the following:
1. 2x – 10
2(x – 5)
2. 3x + 9
3(x + 3)
3. -2x – 10
-2(x + 5)
4. -3x + 9
-3(x – 3)
5. x2 + 3x
x(x + 3)
6. x2 – x
x(x – 1)
7. 2x2 + 2x
2x(x + 1)
8. 3x2 – 9x
3x(x – 3)
Type2: Factoring Difference of 2 Squares
Remember FOIL = First, Outside, Inside, Last
Example1
FOIL: (x + 5)(x – 5)
x2 – 5x + 5x – 25
x2 – 25
Note: FOIL is also the
Double Distributive Property!
Example2
FOIL: (3x – 4)(3x + 4)
9x2 – 12x + 12x – 16
9x2 – 16
Notice that in the above examples the OI terms in FOIL always cancel each other out
because the only change between the 2 binomials is the sign.
One is + ; the other is – .
Factoring the Difference of Two Squares:
Since (a + b)(a – b) = a2 – b2 then
a2 – b2 must be equal to (a + b)(a – b)
How to recognize the Difference of 2 Squares:
= It is a binomial (2 terms)
= The terms have a minus sign between them (Unless they are reversed!!)
= Each term is a perfect square (Ex.16, x2, 25x4)
Here are examples of differences of 2 squares factored:
x2 – 49 = (x + 7)(x – 7)
x2 – y4 = (x + y2)(x – y2)
-25 + x2 = x2 – 25 = (x + 5)(x – 5)
Watch for the SUM of 2 Squares. These are PRIME.
x2 + y2 can’t factor, this is prime
9x2 + 1 can’t factor, this is prime
Practice
Factor the following: (Watch for GCF first, then Diff of 2 Squares!!)
1. x2 – 100
(x + 10)(x – 10)
2. y2 – 1
(y + 1)(y – 1)
3. 2x – 6
2(x – 3)
4. 9x2 – 4
(3x + 2)(3x – 2)
5. 49x2 – 9
(7x + 3)(7x – 3)
6. x2 – 2x
x(x – 2)
7. x2 – x
x(x – 1)
8. 3x3 – 27x
3x(x2 – 9)
3x(x + 3)(x – 3)
9. -9 + x2
x2 – 9
(x + 3)(x – 3)
10. x2 + 9
prime
Type3: Factoring Trinomials (a=1)
The general form of a trinomial is ax2 + bx + c, so if a =1 then these trinomials look
like this: x2 + bx + c, where b and c are positive or negative numbers.
Examples of FOIL
FOIL: (x + 2)(x + 4) =
FOIL: (x – 2)(x – 4) =
FOIL: (x + 7)(x – 3) =
FOIL: (x – 7)(x + 3) =
x2 + 6x + 8
x2 – 6x + 8
x2 + 4x – 21
x2 – 4x – 21
From the above we can make this chart for Factoring trinomials:
x2 + □x + □ = (x + __)(x + __)
x2 – □x + □ = (x – __)(x – __)
x2 + □x – □ = (x + __)(x – __)
x2 – □x – □ = (x – __)(x + __)
Factoring Trinomials
x2 + 8x + 12 = (x + __)(x + __)
x2 – 8x + 12 = (x – __)(x – __)
x2 + 2x – 15 = (x + __)(x – __)
x2 – 2x – 15 = (x – __)(x + __)
Note: If the last term is
pos, signs are the same!
Note: If the last term is
neg, signs are opposite!
Here we need 2 numbers
that multiply to give us
pos 12 and add up to 8.
Here we need 2 numbers
that multiply to give us
neg 15 and add up to 2.
Watch the signs!
Practice
Factor the following: (Watch for GCF first, then Diff of 2 Squares, then Trinomials!!)
1. x2 + 7x + 12
(x + 4)(x + 3)
2. y2 – 49
(y + 7)(y – 7)
3. 2x + 4
2(x + 2)
4. x2 + 2x – 15
(x + 5)(x – 3)
5. 2x2 – 6x
2x(x – 3)
6. x2 – 3x – 28
(x – 7)(x + 4)
7. x2 + x – 56
(x + 8)(x – 7)
8. x2 + 10x + 21
(x + 7)(x + 3)
9. 2x2 + 14x + 20
2(x2 + 7x + 10)
2(x + 5)(x + 2)
10. 3x2 – 300
11. x2 + 10x + 25
2
3(x – 100)
(x + 5)(x + 5)
3(x + 10)(x – 10) or (x + 5)2
12. x2 + 2x + 15
prime
Type 4: Factoring Trinomials (a≠1)
Remember that the general form of a trinomial is ax2 + bx + c
Examples of FOIL with this type
(3x + 2)(2x + 1) = 6x2 + 3x + 4x + 2 = 6x2 + 7x + 2
(3x – 2)(2x – 1) = 6x2 – 3x – 4x + 2 = 6x2 – 7x + 2
(5x + 7)(2x – 3) = 10x2 – 15x + 14x – 21 = 10x2 – x – 21
(5x – 7)(2x + 3) = 10x2 + 15x – 14x – 21 = 10x2 + x – 21
Remember the following patterns from Section 2.5:
x2 + □x + □ = (x + __)(x + __)
x – □x + □ = (x – __)(x – __)
2
x2 + □x – □ = (x + __)(x – __)
x – □x – □ = (x – __)(x + __)
2
Note: If the last term is
pos, signs are the same!
Note: If the last term is
neg, signs are opposite!
Factoring Trinomials Using Intelligent Guessing Method
(We use this method with the 1st and last terms have very few factors, especially if the
numbers are prime!)
2x2 – 7x + 5 = (__x – __ )(__x – __) Write pattern of signs
= (2x – __ )( x – __ ) Choose factors of 1st term.
= (2x – 1) ( x – 5) Choose factors of last term.
Quickly Foil to check this guess. If it doesn’t work, try another guess by
rearranging the factors of the last term. In this case the middle term will become 11x which is not correct!
= (2x – 5) ( x – 1)
Quickly Foil again to check. If it works, then you are done. In this case the middle
term will become -7x which is correct!
Factor the following using the Guess Method:
2x2 + 5x + 2 = (2x + 1 )(x + 2 )
3x2 + 4x – 7 = (3x + 7 ) ( x – 1 )
3x2 – x – 2 = (3x + 2 )(x – 1)
Notice that the signs have to be positive!
Type4 continued: A 2nd Method for Factoring Trinomials where a≠1
If you don’t like the Guess Method, a 2nd way is to use the Box Method to factor.
Before we use the Box Method to Factor, we must know how to use the Box Method to
FOIL or Distribute.
Ex: (2x + 3)(3x + 4)
3x
+4
2x
Step 1. Organize the factors around a 2 by 2 box.
+3
3x
+4
2x
6x2
8x
+3
9x
12
Step 2. Distribute (4 multiplications).
Cross Multiply the values in the
box. What do you notice?
This is very important to see as
we use this idea when factoring
with the box method!
6 · 12 = 72
2
Answer: 6x + 17x + 12
8 · 9 = 72
Step 3. Combine the like terms.
Also notice that 8 + 9 = 17
which is the middle term of the
final answer!
Practice
Multiply the following: (Use the Box Method!)
1. (2x + 3)(x + 5)
2x2 + 13x + 15
2. (3x – 4)(x – 3)
3x2 – 13x + 12
3. (2x + 1)(x – 6)
2x2 – 11x – 6
4. (3x – 5)(2x + 3)
6x2 – x – 15
We can now use the Box Method to Factor the more difficult types:
Ex1: Factor 3x2 + 16x + 20
3x2
___x
Step 1. Fill in the 1st and last terms into the boxes shown.
Notice that the other 2 boxes will both have “x” terms.
___x
20
Note: 3 · 20 = 60 We need this for step 2!
3x2
x
10x
6x
20
3x
+10
3x2
Step 2. Fill in the final 2 boxes with 2 numbers that
multiply to 60. Remember that when you Cross Multiply,
each must be the same result.
These numbers must also add up to 16. (Middle Term)
6 · 10 = 60
6 + 10 = 16
10x
Step 3. Now factor out the GCF of each row and each
column to product the final answer.
+2
6x
20
The final answer is 3x2 + 16x + 20 = (3x + 10)(x + 2)
Check by using FOIL to confirm your answer!
Try the following 3 examples (Note: All signs are positive for these 3 problems.)
#1. 15x2 + 16x + 4
(3x + 2)(5x + 2)
#2. 6x2 + 11x + 4
(2x + 1)(3x + 4)
#3. 6x2 + 11x + 5
(6x + 5)(x + 1)
We can use the Box Method to Factor any of the more difficult types (even those with
negative values): {KEY NOTE: There cannot be any GCF factors in the original
question for this method to work!!}
Ex2: Factor 2x2 – x – 15
2x2
___x
Step 1. Fill in the 1st and last terms into the boxes shown.
Notice that the other 2 boxes will both have “x” terms.
___x
-15
Note: 2 · -15 = -30 We need this for step 2!
2x
+5
2x2
-6x
5x
-15
x
-3
2x2
5x
Step 2. Fill in the final 2 boxes with 2 numbers that
multiply to -30. Remember that when you Cross
Multiply, each must be the same result.
These numbers must also add up to -1. (Middle Term)
5 · -6 = -30
5 + -6 = -1
-6x
-15
Step 3. Now factor out the GCF of each row and each
column to product the final answer. If the first number is
negative in that row or column, factor out the negative also!
The final answer is 2x2 – x – 15= (x – 3)(2x + 5)
Check by using FOIL to confirm your answer!
Practice
Factor the following: (Watch for GCF first, then Diff of 2 Squares, then Trinomials!!)
1. x2 – 4
(x + 2)(x – 2)
2. x2 – x – 6
(x – 3)(x + 2)
3. 6x2 + 7x – 2
prime
4. 2x2 – 6x – 56
2(x2 – 3x – 28)
2(x – 7)(x + 4)
5. 8x2 + 2x – 3
(4x + 3)(2x – 1)
6. 3x2 – 6x
3x(x – 2)
7. 2x2 – 13x + 15
(2x – 3)(x – 5)
8. 6x2 – 13x + 6
(2x – 3)(3x – 2)
Math PreCalc 20 Extra Practice Factoring
(Answers on the next page!)
Factor the following.
(Watch for all the different types!)
1.
5x² + 17x + 12
2. 3x² - 16x + 21
3.
x² - 36
4. 10x² - 17x + 3
5.
3x² + 19x + 30
6. 2x² - 5x - 25
7.
9x² - 4
8. 3x² + 8x - 11
9.
x² + 6x + 8
10. 2x² - 15x + 28
Solutions to previous questions:
1.
5x² + 17x + 12
(5x + 12)(x + 1)
2. 3x² - 16x + 21
(3x - 7)(x - 3)
3.
x² - 36
(x - 6)(x + 6)
4. 10x² - 17x + 3
(5x - 1)(2x - 3)
5.
3x² + 19x + 30
(x + 3)(3x + 10)
6. 2x² - 5x - 25
(2x + 5)(x - 5)
7.
9x² - 4
(3x + 2)(3x - 2)
8. 3x² + 8x - 11
(x - 1)(3x + 11)
9.
x² + 6x + 8
(x + 4)(x + 2)
10. 2x² - 15x + 28
(x - 4)(2x - 7)
Box Method Solutions for some of the above:
5x2 + 17x + 12
#1.
3x2 – 16x + 21
#2.
x
+1
5x
5x2
5x
x
+12
12x
12
-3
10x2 – 17x – 3
#4.
5x
3x
3x2
-9x
-7
-7x
21
2x2 – 5x – 25
#6.
2x
-1
+5
2x
10x2
-2x
x
2x2
5x
-3
-15x
-3
-5
-10x
-25