Algebra
Chapter 8: Factoring Polynomials
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Teacher:____________________________
Pd: _______
Table of Contents
o Day 1: SWBAT: Factor polynomials by using the GCF.
Pgs: 1-6
HW: Pages 7-8
2
o Day 2: SWBAT: Factor quadratic trinomials of the form x + bx + c. (a = 1)
Pgs: 9-13
HW: Page 14
o Day 3: SWBAT: Factor a Difference of Two Squares. “D.O.T.S”
Pgs: 15-17
HW: Page 18-19
o Day 4: SWBAT: Factor a Polynomial Completely
Pgs: 20-23
HW: Pages 24-25
o Day 5: SWBAT: “More” Factoring a Polynomial Completely
Pgs: 26-29
HW: Page 30
o Day 6-7: SWBAT: “Stations “ Review of Factoring
Pgs: 31-33
HW: Pages 34-37
Day 1: Factoring by GCF
SWBAT: Factor polynomials by using the GCF.
Warm – Up
Multiply each polynomial
1.
2.
2x(3
2x + 1)
4xy (3x + 6y - 7)
Recall!
The ___________________ Property: ab + ac = a(b + c).
-
1
Steps to Factoring by GCF
Step 1: Find largest number that divides into ALL terms.
Step 2: Find variables that appear in ALL terms and pull out the smallest exponent for that variable.
Step 3: Write terms as products using the GCF as a factor.
Step 4: Use the Distributive Property to factor out the GCF.
Step 5: Multiply to check your answer. The product is the original polynomial.
Example 1: Factoring by Using the GCF
Factor each polynomial and check your answer.
a) 2x2 + 4
b) 6x2 - 9x
Practice: Factor each polynomial using the GCF and check your answer.
1. 7x + 21
2. 24c2 + 36c
3. 44n3 + 11n2
4. 12x5 – 18x
5. 10g3 – 30g
6. 9m2 + 18m
2
Example 2: Factoring by Using the GCF
Factor each polynomial using the GCF and check your answer.
c) 7n3 + 14n + 21n2
d) 8x4 + 4x3 – 2x2
Practice: Factor each polynomial using the GCF and check your answer.
7. 12h4 + 18h2 – 6h
8. 36f + 18f2 + 3
9. 6n6 + 18n4 – 24n
Example 3: Factoring a common binomial factor Using the GCF
e) 4x(x + 1) + 7(x + 1)
f) y(y – 2) - (y – 2)
Practice: Factor each polynomial and check your answer.
10)
11)
12)
3
Example 4: Factor by Grouping
g)
4
13)
14)
15)
16)
5
Challenge Problem: Factor.
12a2bc2 - 24a4c
Summary:
Exit Ticket:
1)
2)
6
Day 1: Homework:
Factor using the GCF.
1) x 2  x 
2) 6x 2  27 x 
3) 4x 2  10 x 
4) 25x 2 10 x
5) 5x 2  10 x  25 
6) 8x 2  4x  16 
7) 2x3  10 x 2  20 x 
8) 8x 4  4x3  16 x 2 
9) 15x3  30 x 2  45x 
10) 4xy  2x 2 y 2 
11) 45xy2  9x 2 y 
12) 3x 2  6x 
7
13) 12 x 2  3xy 
14) 3xy2  66 y 
15) 6ab  42a 
16) 18x 3  9x 2 
17) 4x3  8x 
18) x 2  2x 
19) 7 xy  21x 2 y 
20)
21) 2k 2t 3  4k 4t 
22)
23)
24)
8
Day 2: Factoring x2 + bx + c
SWBAT: Factor quadratic trinomials of the form x2 + bx + c.
Warm – Up
1. Factor by Grouping.
2. Factor using GCF.
Mini-Lesson
Do you recognize the pattern???
___________________________________________________________________________________
___________________________________________________________________________________
You Try!!! Complete the “Diamond”
Multiply
(x + 2)(x + 5) = _____________________________ = ___________________
 Notice the constant term in the trinomial; it is the product of the constants in the binomials.
 You can use this fact to factor a trinomial into its binomial factors.
(Find two factors of c that add up to b)
9
ax2 + bx + c
Example 1: First Sign is Positive and Last Sign is Positive
Factor: x2 + 6x + 8
Answer: (
Factor: x2 + 5x
)(
)
Answer: (
)(
)
Practice 1: Factor.
1. x2 + 5x + 6
Answer: (
4. x2 + 6x + 9
2. x2 + 8x + 12
)(
)
Answer: (
)(
5. x2 + 10x + 21
3. x2 + 6x + 5
)
Answer: (
)(
)
6. x2 + 11x
10
Example 2: First Sign is Negative and Last Sign is Positive
Factor: x2 - 10x + 24
Answer: (
Factor: x2 - 7x
)(
)
Answer: (
)(
)
Practice 2: Factor.
7. x2 - 8x + 15
Answer: (
8. x2 - 6x + 8
)(
)
Answer: (
10. x2 - 5x + 6
)(
11. x2 - 13x + 40
9. x2 - 7x + 10
)
Answer: (
)(
)
12. x2 - 6x
Example 3: First Sign is Positive or Negative and Last Sign is Negative
Factor: x2 + x - 20
Answer: (
)(
)
11
Practice 3: Factor.
13. x2 + 2x – 15
Answer: (
)(
14. x2 + 3x – 10
)
16. x2 - 2x – 3
Answer: (
17. x2 - 2x – 15
15. x2 + 6x - 40
)(
)
Answer: (
)(
)
18. x2 - 2x - 48
Challenge Problem:
1)
2)Factor: x4 + 18x2 + 81
12
Summary:
Example: Factor: x2 – 5x - 50
Exit Ticket:
13
Day 2: Homework:
Factor each trinomial.
+ 13x
- 9x
- 12x
14
Day 3: Factoring Special Products
SWBAT : Factor a Difference of Two Squares
Warm – Up
The area of the rectangle below is represented by the polynomial x2 + 8x + 7.
Find the binomials that could represent the lengths and width of the rectangle.
A = x2 + 8x + 7
Make a list of perfect squares.
15
Example 1: Factoring the Difference of Two Squares
Factor:
x2 - 25
Practice: Factoring the Difference of Two Squares
Factor.
1) x2 – 64
2) x2 - 9
3) x2 - 81
4) x2 – 100
5) 49 - x2
6) x2 - 81
7) x2 – 1
8) 4 - x2
9) x2 - 121
Example 2: Factoring the Difference of Two Squares
Factor: 64x2 – 1
Factor: x6 - 25
Practice: Factoring the Difference of Two Squares
10) 9x2 – 4
11) 9 - 16x2
12) 49x2 - 64
13) 25x2 – 1
14) x2 - 25y2
15) 16x2 – 25y2
16) 64x2 – 9y2
17) x4 – y10
18) 49x2 – 121y2
16
Challenge Problem
Factor:
1
4
x2
-
1
9
Summary
Exit Ticket:
17
Day 3: Homework - Factoring the Difference of Two Perfect Squares
1. x2 – 36
2. x2 – 1
3. x2 – 25
4. 4x2 – 9
5. x2 – 81
6. 25x2 – 4
7. x2 – y2
8. 64x2 – 25b2
9. x2 – 100
10. x2 – 225
11. x4 – 64
12. x2 – 169
13. 16x2 – 81
14. x6 – 81
15. x2 – 49
.
18
Factor by Grouping.
15.
16.
17.
18.
19
Day 4: Factoring Completely
SWBAT: Factor a Trinomial Completely
Warm – Up
Factor each.
1.
2.
3.
4.
5. Factor by Grouping.
20
Factoring Trinomials Completely
In the previous lesson, we saw how to factor a trinomial of the form bx c by employing the
“diamond” method. In each of those cases, the coefficient of the quadratic ( ) term was
always one, and thus not written. It is also possible to factor trinomials of the form a bx c where
the coefficient a is a number other than 1 by combining two factoring methods into the same problem.
21
22
Challenge Problem:
Recall that the volume of a rectangular solid (a box) is given by V L W H . If a particular
rectangular solid has a volume of 5 15x 10 , how would you represent the length, width and height of the
solid? Justify your answer.
SUMMARY
Exit Ticket
23
Day 4 – Factoring Trinomials Completely Homework
24
25
Day 5: “More“ Factoring Completely
Warm - Up
1.
2.
Some polynomials cannot be factored into the product of two binomials with integer coefficients,
(such as x2 + 16), and are referred to as prime.
Other polynomials contain a multitude of factors.
"Factoring completely" means to continue factoring until no further factors can be found. More
specifically, it means to continue factoring until all factors other than monomial factors are prime
factors. You will have to look at the problems very carefully to be sure that you have found all of
the possible factors.
To factor completely:
1. Search for a greatest common factor. If you
find one, factor it out of the polynomial.
2. Examine what remains, looking for a trinomial
or a binomial which can be factored.
3. Express the answer as the product of all of the
factors you have found.
26
Example 1: Factoring Completely
FACTOR:
10x2 - 40
Practice: Factoring Completely
27
Example 2: Factoring Completely
Factor: 8
Factor: 2
Practice: Factoring Completely
4. 10
5. 2
6.
7.
8.
9.
3. 4x2 + 24x + 36
10.4. x3 - 8x2 + 16x
28
Challenge Problem:
Summary:
Exit Ticket:
29
Day 5: Homework
30
“REVIEW FOR TEST”
SWBAT: Apply their knowledge on Factoring
Station # 1
Common Monomial Factors (GCF)
Factor.
1) 9x2 – 21x5
Factor by Grouping.
4)
2) 4x3 – 6x2 + 10x
3)
5)
Station # 2
Difference of Two Squares “D.O.T.S”
Factor.
1) x² - 49
2) 36x² - y²
3) 64 - y²
4) 9a² - 121y²
5) a6 – 9b12
6) 25x4 - 144y²
31
Station # 3
Factoring Trinomials “Diamond”
1) x² + 21x + 20
2) x² - 10x + 24
3) x² + 3x – 18
4) x² - 7x + 12
5) x² - 6x - 27
6) x² - x – 56
Station # 4
Factoring Completely
1) ax² - a
2) 4a2 – 36
3) 12x2 – 3y²
4) 9a4 – 36b4
5) 3x2 + 15x – 42
6) x4 – 3x3 – 40x²
32
Station # 5
Word Problems
1) The area of rectangle is represented by x2 + 9x + 18. Find the binomials that could represent the
lengths and width of the rectangle.
2) The Volume of rectangular prism is represented by p3 - 12p2 + 35p. Find the factors that would
represent the length, width, and height of the rectangular prism.
33
Chapter 8 Review
SWBAT: Apply Their Knowledge on Factoring.
A)
B)
C)
D)
24t
3t6
t2
t6
A)
B)
C)
D)
2y4
2y2
y3
2y
A)
B)
C)
D)
5n9
3n4
15n
3n9
4.
Factor each expression using the GCF.
5.
6.
7.
8.
34
9.
A)
B)
C)
D)
10.
(x + 6)(x + 1)
(x + 5)(x + 1)
(x - 5)(x + 1)
(x + 2)(x + 3)
11.
A)
B)
C)
D)
(x - 3)(x - 7)
(x - 3)(x + 7)
(x + 10)(x + 11)
(x + 3)(x - 7)
A) (x + 5)(x + 10)
B) (x – 5)(x – 3)
C) (x + 5)(x + 3)
D) (x – 5)( x + 3)
Factor each binomial.
12.
A)
B)
C)
D)
13.
(b - 8)(b - 2)
(b + 4)(b + 4)
(b + 8)(b + 2)
(b - 4)(b + 4)
15.
A)
B)
C)
D)
A)
B)
C)
D)
25x2 – 4
14.
(5x + 2)(5x - 2)
(15x + 2)(10x - 2)
(x + 2)(5x - 2)
(5x + 2)(5x + 2)
16.
3
2
3x (x - 9)
3x3(x + 3)(x - 3)
3x3(x + 3)(x + 3)
9x3(x2 - 9)
17.
35
18.
19.
20.
36
21.
22. A box has a volume given by the trinomial
Use factoring completely.
a.
b.
c.
d.
–
– –
+ 3
–
. What are the possible dimensions of the box?
– –
–
–
–
–
37