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Name ___________________
Integrated Algebra 1A
Factoring
Factoring thought Process:
1. __________________________
2. __________________________
3. __________________________
4. ___________________________
Integrated Algebra 1 A
Day 1 - Factoring and GCF
Prime factorization:
Numbers can also be written as a product of their prime factors.
Recall: Prime Number definition:
_________________________________________________________________________
_________________________________________________________________________
List the following numbers as a product of prime factors: (Use exponential form.)
1. 16
2. 70
3. 72
4. 81
5.) Can you write the following numbers as a product of only prime factors?
(Use exponential form!)
a.) 100
b.) 36
c.) 30
Ex:
So GCF of 15, 30 and 105 =
Example:
Find the GCF of the following monomials:
1.
36xyz and 48xyz
2.
60r2s4 and 36rs2t
3.
24a3b and 18a2bc and 12ac2
4.
42ab2c3
5.
40x2 and 50x3 and 60x7
and 35ac2
and
14abc
The product of two numbers is 24. Find the second factor, if the first factor is:
1.
4
2.
3
3.
12
Example:
is:
The product of two monomials is 81c9d12e. Find the second factor, if the first factor
1.
27e
2.
c2d11
3.
3cd3e
4.
9c4d4
5.
81c5d9e
IA1A Factoring Day 1 HW - Factoring and GCF
1. Using exponential form, write the following numbers as a product of only prime factors.
a.) 60
d.) 86
c.) 32
b.) 44
e)
120
f.) 80
2. Find the greatest common factor for each of the following pairs of monomials.
(d) 30g2 and 15g3
(e) 32x2 and 48x4
(f) 14a4b2 and 21a3b3
(g) 8b5 and 12b3
(h) 50x2y6 and 60x4y9
(i) 20x4y5 and 35x3y7
Integrated Algebra 1 A
Day 2 - GCF Factoring
WARM UP
Complete the puzzle using the single digits: 1, 2, 3, 4, 5, 6, 7, 8 and 9
18 8
35
42
Each digit must be a factor of
the number in the row and
column that the digit appears
in. Each digit is used.
60
72
EXAMPLES
1. 2a + 2b
2. 7y - 7
3. 6y4 + 3y2
4.
15xy2 + 25x2y
5. πr2 + 2πrh
6. 5x2 + 11x
7. 14x - 7x2
8.
3a - 3b
Factor out the GCF
Check
PRACTICE
9.
10.
11.
Factor out the GCF
Check
2x - 4x3
9ab2 - 6ab - 3a
p + prt
12. 21r2s3 - 14r2s
13.
3y7 - 6y5 + 12y3 + 21y2
14. If the area of a rectangle is 14x3 - 21x2 and the width is 7x2, what is the length?
15. If the area of a rectangle is 45h4 - 18h2 and the length is 5h2 - 2, what is the width?
More practice: Factor out the GCF. Write the expression as the product of the GCF and the
remaining factor.
IA1A Factoring Day 2 HW - GCF Factoring
1.
2.
3.
4.
1.
8.
2.
9.
3.
10.
4.
11.
5.
12.
6.
13.
7.
14.
Multiple Choice
Integrated Algebra 1 A
Day 3 - DOPS Factoring
WARM-­‐UP: Multiply
1.
(6 - x)(6 + x)
3.
(a + 2)(a - 2)
2.
(x - 5)(x + 5)
4.
(2x + 1)(2x - 1)
Perfect Squares - Recall!
NUMBERS: A Perfect square is a number that has a whole number square root.
For example, 25 is a perfect square, because 5 (it's square root) is a whole number.
List the first 12 perfect squares:
__________________________________________
VARIABLES: Any variable raised to an even power is a perfect square.
Examples: ____________________
Difference of Perfect square factoring;
Look at the products from the warm up above: What patterns are true about them in all cases?
1.________________________________________________________________
2.________________________________________________________________
3.________________________________________________________________
These special products are called ______________________________________
DOPS?
x2 – 2
a2 – 4
x2 - y2
2x2 - 49
4x2 – 1
x4 - 25y2
x2 + 9
9a2 + 64
1. Create TWO sets of parenthesis. One Plus, one minus.
2. Take the square root of the first term, put it in the first spot of the binomial.
3. Take the square root of the last term, put it in the last spot of the binomial.
FACTOR:
x2 – 25
16x2 - 49
a2 – 4
x2 - y2
4x2 – 1
x4 - 25y2
Factor using DOPS Factoring:
(g) x2 - 9
(j) 16 - y2
(h) c2 - 49
(k)d4
- 121
m.) Express the area of each shaded region as:
a) the difference of the areas shown
b) the product of two binomials
(i) 100 - y2
(l) x4 - 25
IA1A Factoring
Day 3 HW - DOPS Factoring
Factor using the difference of perfect square factoring:
17.
Integrated Algebra 1 A
Day 4 - Factor Completely So Far
Warm Up
1. If 391 can be written as 400 - 9, find two factors of 391 without
using a calculator. Explain.
2. Factor:
4x2 + 9
3. Factor: 8x2y7 - 12x4y8
4. Factor: 81x2 - 16
Factor Completely
There will almost always be more than one step in the factoring process.
You need to have a thought process, about which factoring method to
use first.
The Factoring Thought Process so far:
1.Look for GCF.
2.If there is a GCF, factor it out.
3.Now look at the remaining factor.
4.Check for DOPs. If you have DOPS, do DOPs factoring.
Example:
Factor completely: 6x2 - 54
Factor Completely:
Ex: 125 - 5x2
Ex: 4ax2 - 36a
Ex: 10x2 - 1000
Ex: 5x2y - 320y
Ex: 18 - 2a2
Ex: 3x6 - 12
Ex: 3y4 - 3
Ex: 16 - 2b2
Ex: 7cd2 + 28c
Ex:
Ex: 3ab6 - 48a
Ex: 16 - 4x4
Ex: 4x4 - 100
Ex: 2cd2 - 72c
IA1A Factoring
Day 4 HW - Factor completely so Far
1. 3a2 - 48
2. 2y2 - 50
3. 125 - 5x2
4. 2x2 - 72
5. 6x2 - 6
6. 5r2 - 45
7. x3 - 9x
8. x5 - 4x3
9. 128 - 2x2
10. 3y2 - 27
11. 2xy2 - 8x
12. 2x2 - 98
13. 3xy2 - 48x
14. 20x3 - 80x
15.
Integrated Algebra 1 A
Day 5 - Factor Completely So Far - Day 2
Factor Puzzles and Quiz 1 Today
Warm Up
Factor completely. Completely!!!!! Show both steps!
1. 3x2 - 48
2. 18 - 2y6
4. 27 - 3r4
5. 7x2y - 28y
3. x3 - 9x
6. 5r5 - 25
Today,
Factoring puzzle - on separate sheet,
followed by quiz 1.
Quiz 1: Factoring with GCF and DOPS
IA1A Factoring
Day 5 HW - Mixed factoring Practice
Factor, by using GCF factoring.
Factor, using difference of perfect square factoring.
Key to HW 5
Integrated Algebra 1 A
Day 6 - Trinomial Factoring
Warm-up Review
1. FOIL
3. MULTIPLY
(x + 13)(x - 13)
5x2(2x - 1)
2. FACTOR
4. FACTOR
Name the two types of factoring you know so far!
Factoring Trinomials
x2 - 100
4x2 - 12x
Think about these:
Find me two numbers that.......
10
16
Mult
Add
11
30
Add
Mult
2
Add
12
Add
1
63
2
Subtract
to
Subtract
to
Mult
24
14
15
Mult
Mult
20
Mult
42
Subtract
to
3
Subtract
to
Mult
28
Mult
We need to use this thought process as we factor trinomials. Find two numbers that multiply to
the last term of the binomial and add or subtract to the middle term.
See you need factors of 12 that have a sum of 8 for this factoring to be correct.
Steps in factoring trinomials:
1.) Set up two sets of parenthesis.
2.) Look at the constant term (the last term) to determine the sign. If positive signs are same. If
negative signs are different.
3.) If signs are the same - look at the linear term (middle term) to determine what the signs are.
When signs are same - add for the middle.
( "+" in middle: both plus, "-" in middle: both minus)
4.) If signs are different - subtract for the middle.
(If middle term is negative, bigger factor is negative, middle term pos, bigger factor positive.)
5.) Find the right combination of factors that multiply out for the last term and add or subtract for
the middle term.
6.) FOIL OUT to check!!!
Examples, Factor:
Review the factoring thought process so far:
1.)_________________2.)___________________ 3.)____________________
IA1A Factoring
Day 6 HW - Trinomial Factoring
Factor, using trinomial factoring.
Integrated Algebra 1 A
Day 7 - Trinomial Factoring - D2
Warm Up - Factor completely:
Ex:2x2 - 18
Factoring Trinomials
Ex:4x5 - 100x3
Ex: 3x2 + 27
ax2 + bx + c
The first term you always check when you begin factoring is ____ ?
Why?
If the sign of the last term is positive, the signs in the binomials are?
If the sign of the last term is negative the signs in the binomials are?
When the signs in the binomials are the same, you _______ for the
middle term.
When the signs in the binomials are different, you ___________ for
the middle term.
Find two numbers that:
-Multiply to 15 and add to 8
-Multiply to 36 and add to 12.
-Multiply to 27 and add to 12
-Multiply to 45 and add to 14.
-Multiply to 54 and add to 15
-Multiply to 72 and add to 17.
-Multiply to 63 and add to 16
-Multiply to 42 and add to 13.
Find two numbers that:
-Multiply to 15 and have a difference of 2
-Multiply to 16 and have a difference of 6
-Multiply to 42 and have a difference of 11
-Multiply to 35 and have a difference of 2
-Multiply to 48 and have a difference of 2
-Multiply to 50 and have a difference of 5
-Multiply to 28 and have a difference of 3
-Multiply to 56 and have a difference of 1
More practice factoring Trinomials:
1.) x2 + 8x + 15
8.) x2 + 8x + 16
2.) t2 - 7t - 60
9.) x2 + 5x + 4
3.) x2 + 5x - 21
10.) x2 + 7x + 12
4.) y2 - 9y + 8
11.) x2 - 9x + 20
5.) x2 - 10x + 21
12.) x2 + 8x + 12
6.) y2 - 2y + 1
13.) x2 - 6x + 9
7.) x4 - 3x2 - 10
14.) x2 + 2x - 15
More Trinomial Factoring Practice:
15.) x2 + 10x + 21
16.) x2 + 3x - 10
17.) x2 + 5x - 24
18.) x2 - 7x - 18
19.) x2 + 9x + 18
20.) a2 - 15a +56
21.) r2 - 2r - 15
22.) x2 + 2x + 1
23.) x2 - 4x - 21
24.) x2 + 2x - 63
25.) x2 + 10x + 24
26.) y2 - 9y + 20
27.) f2 - 2f - 63
28.) w2 + 7w + 12
IA1A Factoring
Day 7 HW - Trinomial Factoring
Factor using trinomial factoring:
Integrated Algebra 1 A
Day 8 - Trinomial Factoring - D3
Factor:
Warm Up
1.
32x5y4 - 24x4y4
3. 16 - y2
5. x2 - x - 20
2.
12x6 - 10x5 - 2x2
4.x2 - 100
6. x2 - 6x + 8
Today: Tiling Worksheet Activity:
Using the digits from 0 - 9, fill in the squares with the appropriate digit.
1 2 3 4 5 6 7 8 9 0
1 2 3 4 5 6 7 8 9 0
1
3
4 5 6 7 8 9 0
2
1
2 3 4 5 6 7 8 9 0
IA1A Factoring
Day 8 Classwork and Homework - Trinomial Factoring
Begin in class and complete for Homework: Factor using trinomial factoring:
Write prime - if not factorable.
Integrated Algebra 1 A
Day 8a - Factoring Special Cases
Foil:
Warm Up
1.) (x + 3)(x + 3) =
2.) (x - 5)(x - 5) =
3.) (y - 7)2 =
4.) (x + 4)2 =
Multiplying the same binomial by itself, is referred to as the binomial squared!
Factoring Special Cases
Look at the products above, what patterns do you see?
1.
______________
2.
______________
3.
______________
These products are called Perfect square trinomials, when you
see these special products they always factor back into the
binomial squared.
To factor perfect square trinomials:
1. Set up two parenthesis. Both with the sign of the linear (middle) term of
the perfect square trinomial.
2. Take the square root of the first term, put it in the first spot.
3. Take the square root of the last term and put it in the last spot.
4. Both of the binomials will be the exact same - can be written as the
binomial squared!
5 Foil out to check!
Examples, Factor:
x2 - 12x + 36
x2 + 4x + 4
Practice: Factor each of the following perfect square trinomials.
1) a2 + 4a + 4
7) y2 – 20y + 100
2) p2 + 2p + 1
8) 49 + 14a + a2
3)
x2 – 10x + 25
9) 9x2 + 24x + 16
4)
y2 – 8y + 16
10) 16t2 – 40t + 25
5)
r2 + 24r + 144
11) 25k2 – 20k + 4
6) k2 + 121 + 22k
12) 4c2 + 12c + 9
IA1A Factoring
Day 8a Homework - Trinomial Factoring - Perfect Square Trinomials
Directions: Factor as a product of two binomials.
1. x2 + 4x + 4
2. y2 - 12y + 36
3. x2 - 18x + 81
4. u2 + 6u + 9
5. 25 + 10p + p2
6. x2 + 16x + 64
7. y2 - 14y + 49
8. The area of a square is represented by x2 + 18x + 81, what is a side of
the square?
9. The area of a square is represented by 4x2 - 12x + 9, what is a side of
the square?
10. The area of a square is represented by 9x2 + 24x + 16, what is a side of
the square?
Integrated Algebra 1 A
Day 9 - Guess and Check Factoring and Quiz 2.
Warm Up
Practice For Quiz - Factor
1. 35x3y4 - 40x4y5
2.
14x6 - 21x5 - 7x2
3. 25 - a2b2
4. 16a2 - 121
5. x2 - x - 20
6. x2 - 6x + 8
Guess and Check Factoring:
Foil out:
(2x + 3)(x + 5)
Today we will factor trinomials like these products! You
can't just say, you are looking for two numbers that
multiply to the last term and add or subtract for the middle
term - you have to take into account the coefficient in front
of x2!
To factor these trinomials:
1.) we use a methodical guess and check approach, or
2.) Decomposition
Guess and check factoring!!
1.) 3x2 + 8x + 5
2.) 2x2 + 9x - 5
3.) 5x2 - 11x + 2
4.) 2x2 - 11x - 6
Let's review the factoring thought process now that we
have added a new type of factoring:
1.________________
2.________________
3.________________
4.________________
IA1A Factoring
Day 9 Homework - Guess and Check Factoring
1. 2x2 + 5x + 3
6.) 2x2 + 9x - 5
2. 3x2 + 13x + 4
7.) 2x2 - x - 15
3. 2x2 + 11x + 5
8.) 2x2 - x - 6
4. 2x2 + 13x - 7
9.) 3x2 - 7x + 2
5. 3x2 + 5x - 2
10.) 5x2 + 9x - 2
IA1A Factoring
Factor using the guess and check method
6.) 2x2 + 9x - 5
1. 2x2 + 5x + 3
(
)(
)
)(
)
)(
)
)(
)
)(
(
)(
)
(
)(
)
(
)(
)
10.) 5x2 + 9x - 2
5. 3x2 + 5x - 2
(
)
9.) 3x2 - 7x + 2
4. 2x2 + 13x - 7
(
)(
8.) 2x2 - x - 6
3. 2x2 + 11x + 5
(
(
7.) 2x2 - x - 15
2. 3x2 + 13x + 4
(
Last nights HW
Name _______________________
)
(
)(
)
Integrated Algebra 1 A
Day 10 - Factor Completely
Warm Up -
Name the type of factoring and factor.
1. 8r4s4 - 4r5s3
2
.144 - y8
3. x2 - x - 30
4. x2 - 6x + 5
5. 15x5 - 10x3 - 5x2
6.
g6 - 81
The Factoring Thought Process:
1.
______________
2.
______________
3.
______________
4.
______________
Factor completely - means there will be more than
one step, follow the factoring thought process.
Ex:
3x - 9x + 6
2
Remember - One step at a time....
Check for what first?
Factor completely:
1. 5x3 - 45x
2. 2x3y - 10x2y - 28xy
3. 12x + 9x + 6
2
4. 21x2 + 15x + 18
5. 5x - 20x
3
6. 3x5 - 12x3
Remember - One step at a time....
Check for what first?
Factor Completely:
7.) 4x3 - 8x2 - 60x
11.) 4x y + 4xy - 24y
2
8.) 5y3 - 125y
12.) 2x + 2x - 24x
9.) 3x5 - 15x4 + 18
13.) 2x2 + 6x - 36
10.) x4 - 1
4
14.) a8 - 1
3
2
IA1A Factoring
Day 10 Homework - Factor Completely
IA1A Factoring
Day 10 Homework - Factor Completely - page 2
Day 11 - Optional
More Factoring Completely Practice