Algebra 1
Unit 6B – Factoring
Monday Tuesday Wednesday Thursday Friday
9 A Day 10 B Day 11 A Day 12 B Day 13 A Day
Test – Exponents and Polynomials
Factor GCF and Trinomials
− box method
Factoring Trinomials
Feb. 16 17 B Day 18 A Day 19 B Day 20 A Day
No School
Staff Development
Factoring
Trinomials
Factoring with Patterns
− GCF
− difference of squares
− perfect square trinomials
Retest – CBA #4
Dividing Polynomials
Quiz – Factoring
23 B Day 24 A Day 25 B Day 26 A Day 27 B Day
Divide Polynomials
Quiz – Factoring
Elaboration day
Test – CBA #6
(grade will need to go on the NEXT six
weeks marking period)
1
WARM-UP #_______
Simplify each expression.
1. (x + 4)(x – 6) 2. (10x2 + 5x – 6) – (8x
2 – 2x + 7)
3. 5x2 (2xy – 3x) 4. (2x + 5)(3x + 6)
5. (x2 + y
2) – (-x
2 + y
2) 6.
4 2
2 6
a b
a b
2
Notes – GCF and Factoring
Prime Number – a whole number, greater than 1, whose only factors are 1 and itself
Composite Number – a whole number, greater than 1, that is not prime
Prime Factorization – a whole number expressed as a product of factors are all prime
numbers (i.e. factor tree)
Greatest Common Factor (GCF) – the greatest common factor of two or more integers is
the greatest number that is a factor of all the integers
EX1: State whether each number is prime or composite. If the number if composite, find
its prime factorization (tree).
a. 28 b. 61 c. 112 d. 150
EX2: Find the GCF between two numbers using the calculator.
a. -45, 15
b. 169, 13
c. -20, 440
d. 96, 12, -8
Greatest common factor for the same variable will be LOWEST exponent of that given
variable.
Factoring – to express a polynomial as the product of a monomial and a polynomial
EX3: Find the GCF for each set of monomials.
a. x2, x
5, x
4 b. 49x, 343x
2
c. 4a7b, 28ab d. 96y, 12x, -8y
3
Notes – GCF and Factoring
EX4: Factor each polynomial.
a. 24w + 72z b. 30ab2 + a
2b – 12ac
3
c. x4 – 18x
2 + 22x d. a + 10a
2b
3
e. 88x4 – 11x
7 + 66x
5 f. 14c
3 – 42c
5 – 49c
4
g. 48w2x + 18wx
2 – 36wx h. -x
5 – 4x
4 + 23x
3 – x
6
i. 8x – 7y + w j. 18y2 – 50 k. x
3 + 2x
2 + x
4
Reverse Distribution
Find a monomial and a trinomial whose product is equal to each problem below. Cut and paste it in the
correct place.
Problems Monomials
(GCF) Trinomials
1. 12x2+ 3x – 6
2. 12x4 – 6x
2 + 3x
3. 24x5+ 12x
4 – 4x
3
4. 4x4 – 12x
3 + 6x
2
5. 6x3 – 24x
2 – 12x
6. 10a4b
2 – 5a
3b + a
2b
7. 5a6b
5+ 10a
5b
4 – 15a
4b
3
8. 20a5b
5 + 10a
4b
4 – 30a
3b
3
9. 50a6b
2 – 30a
5b
3+ 10a
4b
4
10. 50a7b
6 – 15a
5b
2 + 25a
3
5
Monomials (GCF) Trinomials
6x (10a2b – 5a + 1)
5a4b
3 (10a
4b
6 – 3a
2b
2+ 5)
10a4b
2 (a
2b
2 + 2ab – 3)
5a3 (4x
2 + x – 2)
3 (x2 – 4x – 2)
3x (2x2 – 6x + 3)
a2b (2a
2b
2 + ab – 3)
10a3b
3 (6x
2 + 3x – 1)
4x3 (5a
2 – 3ab + b 2 )
2x2 (4x
3 – 2x + 1)
6
GCF and Factoring
Factor out the GCF.
1. x3 + x
2 + x 2. 15a + 12b + 6c
3. 8x2 – 18y
2 4. x
2y – 2y
5. z3
+ 4z 6. 4x2 – 4x
7. 15x2 – 50x – 10 8. 12a – 11b
9. 64c3 – 56c
2 + 88c 10. 24x
6y
3 – 32x
3y
2 – 20x
2y
2
11. –2x4 + 24x
2 12. 12x
3y
4 – 40xy
5
Name Date
7
Simplify each expression.
13. (2x + 5xy + 7y) + (3x + 7xy + y) 14. (3x + 2y) – (5x + 6y)
15. (a + b)0 16. 95
71
ba20
ba40−−
−−
17. (2m-4
n3)(-5mn
-7) 18.
19. Find the volume of a cylinder with a diameter of 4x3y and a height of 7x
2y
4.
20. Find the volume of a cube with sides 2b3r
2.
Solve.
21. 4x + 2 = 2(5x – 11) 22. 9 – 4x < 10
2
1b
a−
−
8
WARM-UP #_____
Find the missing information on the given rectangles.
What is the area?
This is the same size rectangle just divided up.
What is the area of the first rectangle?
What is the area of the second rectangle?
What is the area of the whole rectangle?
Write the area of each rectangle inside each box for both of the rectangle below and
answer the questions.
What is the total area of the rectangle?
What is the total area of the rectangle?
What do you notice about all of the rectangles above?
What is special about the length and width?
What is the length and width of all the rectangles?
12
17
12
12 5
3
12 5
9
6
15 2
6
9
EXPLORE
Given the rectangles below, determine the length and width of each rectangle and the area. Rectangles
are not drawn to scale.
Total Area ______ Total Area ______
Length _______ Length _______
Width ________ Width ________
Total Area ______ Total Area ______
Length _______ Length _______
Width ________ Width ________
Total Area ______ Total Area ______
Length _______ Length _______
Width ________ Width ________
Total Area ______ Total Area ______
Length _______ Length _______
Width ________ Width ________
Total Area ______
Length __(x – 1)_____
Width __(2x + 3)______
5 2
5
6
14 12
21
20 12
21
2x 5x
25
10
3x2 2x
20
30x
1 2x
4x
-1
20
2
x
30x
x2
-12
-4
x
10
Notes – Factoring Trinomials
EX1. Recall the box method to multiply two binomials. Multiply (x – 3)(x + 2).
Factors:
Product:
EX2. Find the missing dimension of each trinomial’s box. Fill in the blank cells in each box.
a. a2 + 7a + 10 = (a + 5)( ) b. c
2 – 10c + 21 = (c – 3)( )
c. y2 – 2y – 15 = (y + 3) ( ) d. n
2 + 3n – 28 = (n – 4) ( )
e. How do the quantities you filled in the 2 blank cells relate to the original trinomial?
a
5
a2
10
c -3
c2
21
n -4
n2
-28
y
3
y2
-15
11
Notes – Factoring Trinomials
EX3. Write the numbers that give a sum of –5x and a product of –50x2.
• Standard form:
EX4. Factor each trinomial.
a. x2 + 7x + 10 =
Sums to be: (Middle term: b)
Yield a product of:
(this comes from multiplying the a and c)
b. x2 + 3x – 4 =
Sums to be: (Middle term: b)
Yield a product of:
(this comes from multiplying the a and c)
c. x2 – 64 =
Sums to be: (Middle term: b)
Yield a product of:
(this comes from multiplying the a and c)
12
Notes – Factoring Trinomials
d. 3x2 + 14x + 8 =
Sums to be: (Middle term: b)
Yield a product of:
(this comes from multiplying the a and c)
e. 2y2 – 7y + 6 =
Sums to be: (Middle term: b)
Yield a product of:
(this comes from multiplying the a and c)
f. 6x2 – 21x – 12 =
Sums to be: (Middle term: b)
Yield a product of:
(this comes from multiplying the a and c)
13
Notes – Factoring Trinomials
Factoring Using Algebra Tiles
EX5. Determine the factors of each polynomial.
a. b.
c. d.
14
___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
11 15 10 11 1 8 16 7 17 12 5 11 2 16 17 6 4 3
___ ___ ___ ___ ___ ___ ___ ___ ___ ___
14 8 7 9 6 4 3 14 16 13
A B C D E
(x – 2)(5x – 8) (2x + 3)(3x – 2) (x – 13)(x + 3) (x + 2)(x – 3) (x – 2)(x + 3)
F G H I J
(x – 8)(2x + 5) (x + 2)(7x + 3) (x – 2)(x + 1) (x – 5)(x + 3) (x + 5)(x – 3)
K L M N O
(x – 7)(x + 7) (x – 2)(x – 8) (x + 2)(x + 8) (x – 7)(x –2) (x + 7)2
P Q R S T
(x – 7)(x + 2) (x + 2)(x – 8) (x – 3)(x + 3) (x – 6 )(2x – 1) (x + 6)2
U V W X Y
(x – 5)(x + 5) (x – 5)(x – 5) (x – 3)2 (x – 3)(x + 20) (x – 6)(x – 2)
Z
(x + 9)(x + 7)
Factoring Trinomials
Directions: Match that answer to the correct letter of the alphabet. Enter that letter of the alphabet on
the blank corresponding to the problem number.
Factor each polynomial – make sure to show your work.
1. x2 – 49 2. x
2 + 12x + 36 3. 7x
2 + 17x + 6
4. x2 – 9x + 14 5. 5x
2 – 18x + 16 6. x
2 – 2x – 15
Name Date
15
7. x2 – 25 8. x
2 – 8x + 12 9. 2x
2 – 13x + 6
10. x2 + x – 6 11. x
2 – 10x – 39 12. 2x
2 – 11x – 40
13. x2 + 17x – 60 14. 6x
2 + 5x – 6 15. x
2 – x – 2
16. x2 + 14x + 49 17. x
2 – 9
Identify the simplified area of each rectangle. Then determine the factors.
18. 19. 20.
16
WARM-UP #______
1. Factor: x2 + 13x + 12
Factors:
What did you notice?
2. Factor: 4x2 – 9
Factors:
What did you notice?
3. Factor: x2 + 6x + 9
Factors:
What did you notice?
17
18
Explain – Factoring with Patterns
Difference of Squares
a2 – b
2 = (a)
2 – (b)
2 = (a + b)(a – b) Conjugate pairs
difference opposite signs
*Warning: a2 + b
2 does not factor
To recognize perfect squares, look for coefficients that are squares of integers and
variables raised to even powers.
EX1: Factor, if possible, using the difference of squares.
a. 4x2 – 9y
2
b. a2 – 16b
2
c. 9x4 – 25y
4
d. u2v
2 – w
2z
2
e. 25m2 + 36n
2
19
Explain – Factoring with Patterns
Perfect Square Trinomials
• a2 + 2ab + b
2 = (a + b)(a + b) = (a + b)
2
• a2 – 2ab + b
2 = (a – b)(a – b) = (a – b)
2
EX2: Factor each of the following.
a. x2 + 6x + 9
b. x2 – 10x + 25
c. a2 + 8a + 16
d. 9a2 – 24a + 16
20
Factoring Patterns
Determine whether each statement is TRUE. If not, find the correct product.
1. (3x + 1)2 = 9x
2 + 6x + 1 2. (m – 4)
2 = m
2 – 16m + 16
3. (5t – 2)2 = 25t
2 – 20t + 4 4. (2n + 7)
2 = 4n
2 + 28n + 49
5. (2b + 3)2 = 4b
2 + 12b + 6 6. (2a + b)
2 = 4a
2 + 4ab + b
2
Factor each polynomial. If it cannot be factored, write prime.
7. t2 – 12t + 36 8. a
2 + 2ab + b
2
9. 4t2 + 20t + 25 10. n
2 – 1
11. 144 – 25n2 12. 16a
2 – 24ab + 9b
2
Name Date
21
13. t2 – 18t + 81 14. 4n
2 – 9
15. 25 + 10t + t2 16. n
2 – 49
17. 49m2 – 16n
2 18. a
2 – 8a + 64
19. 49a2 + 14a + 1 20. 81 – 121n
2
21. Which is the correct factorization of –45x2 + 20y
2?
A. –5(3x + 2y)2 B. 5(3x – 2y)
2
C. –5(3x + 2y)(3x – 2y) D. 5(3x + 2y)(3x – 2y)
22. Challenge Determine the value(s) of k for which each expression is a perfect square
trinomial.
a. 49x2 – 84k + k b. 4x
2 + kx + 9
22
Explore – Dividing Polynomials
Remember when we MULTIPLIED: (using a box)
(x + 2)(x + 6) or (2y + 1)(3y – 4)
So can you now DIVIDE these polynomials: (using a box)
2x
12x8x2
+
++
43y
4y56y2
−
−−
So…..
2x
12x8x2
+
++
43y
4y56y2
−
−−
the quotient is: the quotient is:
x
+2
–4 3y
23
24
Explain – Dividing Polynomials
Dividing is the opposite operation of . Therefore, we will use
the to assist in dividing trinomials when given a trinomial divided
by a binomial.
EX1. Simplify each expression.
a. 1x
5x4x2
−
−+ Quotient:
b. 4x
12x112x2
+
++ Quotient:
c. 13x
5x13x62
−
−+ Quotient:
x
–1
x2
–x
5x
–5
2x2
3x
8x
12
25
Explain – Dividing Polynomials
EX2. Simplify each expression – ON OUR OWN.
a. 7x
14x33x52
+
−+ Quotient:
b. 8x
24x11x2
−
+− Quotient:
c. 12x
9x17x22
−
−+ Quotient:
26
Gingerbread Man
Divide each polynomial. Each answer determines the next location of the traveling gingerbread man.
Determine the path the gingerbread man makes through the school.
1. 9x
27x12x2
+
++ 2.
8x
40x13x2
−
+− 3.
4x
44x7x2
+
−−
4. 6x
42xx2
−
−+ 5.
1x2
4x9x22
+
++ 6.
7x
7x15x22
−
+−
7. 3x2
15x7x22
−
−+ 8.
5x2
5x17x62
+
++ 9.
7x
14x19x32
+
−+
10. 4x3
12x7x122
+
−+ 11.
1x2
3x2x82
−
−+ 12.
4x3
4x15x92
−
+−
Name Date
27
Principal Secretary
Attendance Counselor
Nurse
Cafeteria
Theater
Playground
2
Library
Home Ec
Lab
Computer
Lab
Art Room
5th
Grade
4th
Grade
3rd
Grade
Tea
cher
Wo
rkro
om
K -
2
Tro
phy c
ase
Kindergarte
n
1st Grade
2nd
Grade
Playground
1
Tea
cher
Wo
rkro
om
3 –
5
Front Door
Where’s my
class? •
(x + 3)
•
(x – 5)
•
(x – 11)
•
(x + 7)
•
(x + 4)
•
(x – 7)
•
(x + 5)
•
(3x + 1)
•
(3x – 2)
•
(4x – 3)
•
(4x + 3)
•
(3x – 1)
•
(x + 11)
•
(x – 4)
•
(3x + 2)
•
(2x – 1)
•
(x + 2) •
(x – 2)
•
(x + 1)
•
(x + 6)
28
Review – CBA #6
1. Find the volume of a cube with sides 2b3r
2.
2. Find the volume of a cylinder that has a radius of 5s3t
5 and a height of 2s
2t
4.
3. Find the area of a triangle that has a base of 32mn7 and a height of 3m
4n
3.
4. If a rectangle has an area of 16x7y
4 and a length of 4x
3y, what is its width?
5. Distance (d), rate (r), and time (t) are related by the formula d = rt. If a ball rolls 36p4q
9 feet for
4p2q
3 minutes, what is the rate?
6. Write an expression that best represents the area of a square with sides of 7x4y
3?
7. Find the perimeter and area of the rectangle in terms of n.
3n – 5
2n + 10
Name Date
29
5n – 1 3n + 5
2n
8. Find the perimeter and area of the triangle in terms of x.
Simplify each expression.
9. (-2x + x2) – x(5x – 4) + (9x
2 – 6x) 10. p(2p – 3) + (p – 3)(4p + 1)
11. (3x5)3(2x
7)2 12. (-3x
6)2
13. (3r + 7)2
14.
8 7
2 6 5
12x y z
4x y z
−
− −
15. ( ) ( )
4 2 3 9 0
10 4
2a b 7ab
3a b−
−
−
16. n6 + n + n
6
17. The dimensions of a wall are 7xy feet by 8x2y
3 feet. A picture has
dimensions 2x feet by x2y
4 feet. If the picture is hanging on the wall as
shown, what is the area of the wall not covered by the picture?
18. A pitcher contains 16x5y
4 ounces of water. A mug holds 2x
2y ounces. Leticia pours water from
the full pitcher into mugs. If she filled axby
c mugs, what is the value of
a + b + c?
30
19. Find the area of a circle with radius 6r3s
5 inches.
20. Find the area of a rectangle with side lengths (x2 – 7x) and (2x
2 + 3x + 1).
21. Describe and correct the error in finding the product of the given polynomials.
22. The area of a rectangle is 3x2 – 10x – 8. Find the dimensions (length and width) of the rectangle.
Factor out the greatest common monomial factor.
23. 16a2 – 40b 24. -36s
3 + 18s
2 – 54s 25. 17abc
2 – 6a
2c
31
Completely factor each of the following polynomials.
26. r2 + 2r – 24 27. y
2 – 2y – 15 28. 2x
2 + 12x + 16
29. 2ax2 – 3ax – 35a 30. 4a
2 + 9a – 9 31. 6k
2 + 13k + 6
32. 9x2 – 121 33. k
2 – 49 34. 64u
2 – 25
Identify the simplified area of each rectangle. Then determine the factors.
35. 36. 37.
32