Algebra - White Plains Middle School 27x 3) 4x2 10x 4) 25x2 10x 5) 5x2 10x 25 6) 8x2 4x 16 7) 2x3...
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Algebra Chapter 8: Factoring Polynomials Name:______________________________ Teacher:____________________________ Pd: _______
Transcript of Algebra - White Plains Middle School 27x 3) 4x2 10x 4) 25x2 10x 5) 5x2 10x 25 6) 8x2 4x 16 7) 2x3...
Algebra
Chapter 8: Factoring Polynomials
Name:______________________________
Teacher:____________________________
Pd: _______
Table of Contents
o Day 1: SWBAT: Factor polynomials by using the GCF. Pgs: 1-6
HW: Pages 7-8
o Day 2: SWBAT: Factor quadratic trinomials of the form x2 + 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
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Day 1: Factoring by GCF SWBAT: Factor polynomials by using the GCF.
Warm – Up Multiply each polynomial
1. 2x(3 2x + 1) 2. 4xy (3x + 6y - 7)
Recall! The ___________________ Property: ab + ac = a(b + c). -
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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 4. 12x5 – 18x
2. 24c2 + 36c 5. 10g3 – 30g
3. 44n3 + 11n2 6. 9m2 + 18m
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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)
4
Example 4: Factor by Grouping
g)
5
13) 14) 15) 16)
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Challenge Problem: Factor.
12a2bc2 - 24a4c
Summary: Exit Ticket:
1)
2)
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Day 1: Homework: Factor using the GCF. 1) xx2 2) xx 276 2 3) xx 104 2
4) xx 1025 2 5) 25105 2 xx 6) 1648 2 xx
7) xxx 20102 23 8) 234 1648 xxx 9) xxx 453015 23
10) 2224 yxxy 11) yxxy 22 945 12) xx 63 2
8
13) xyx 312 2 14) yxy 663 2 15) aab 426
16) 23 918 xx 17) xx 84 3 18) xx 22
19) yxxy 2217 20) 21) tktk 432 42
22) 23) 24)
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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)
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ax2 + bx + c
Example 1: First Sign is Positive and Last Sign is Positive
Factor: x2 + 6x + 8 Factor: x2 + 5x
Answer: ( ) ( ) Answer: ( ) ( ) Practice 1: Factor.
1. x2 + 5x + 6 2. x2 + 8x + 12 3. x2 + 6x + 5
Answer: ( ) ( ) Answer: ( ) ( ) Answer: ( ) ( )
4. x2 + 6x + 9 5. x2 + 10x + 21 6. x2 + 11x
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Example 2: First Sign is Negative and Last Sign is Positive
Factor: x2 - 10x + 24 Factor: x2 - 7x
Answer: ( ) ( ) Answer: ( ) ( ) Practice 2: Factor.
7. x2 - 8x + 15 8. x2 - 6x + 8 9. x2 - 7x + 10
Answer: ( ) ( ) Answer: ( ) ( ) Answer: ( ) ( )
10. x2 - 5x + 6 11. x2 - 13x + 40 12. x2 - 6x
Example 3: First Sign is Positive or Negative and Last Sign is Negative
Factor: x2 + x - 20
Answer: ( ) ( )
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Practice 3: Factor. 13. x2 + 2x – 15 14. x2 + 3x – 10 15. x2 + 6x - 40
Answer: ( ) ( ) Answer: ( ) ( ) Answer: ( ) ( )
16. x2 - 2x – 3 17. x2 - 2x – 15 18. x2 - 2x - 48
Challenge Problem:
1) 2)Factor: x4 + 18x2 + 81
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Summary: Example: Factor: x2 – 5x - 50 Exit Ticket:
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Day 2: Homework: Factor each trinomial.
+ 13x
- 9x
- 12x
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Day 3: Factoring Special Products
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. Make a list of perfect squares.
SWBAT : Factor a Difference of Two Squares
A = x2 + 8x + 7
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Example 1: Factoring the Difference of Two Squares Practice: Factoring the Difference of Two Squares
Factor.
1) x2 – 64 2) x
2 - 9 3) x
2 - 81
4) x2 – 100 5) 49 - x
2 6) x
2 - 81
7) x2 – 1 8) 4 - x
2 9) x
2 - 121
Example 2: Factoring the Difference of Two Squares
Factor: 64x2 – 1
Practice: Factoring the Difference of Two Squares
10) 9x2 – 4 11) 9 - 16x
2 12) 49x
2 - 64
13) 25x2 – 1 14) x
2 - 25y
2 15) 16x
2 – 25y
2
16) 64x2 – 9y
2 17) x
4 – y
10 18) 49x
2 – 121y
2
Factor: x2 - 25
Factor: x6 - 25
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Challenge Problem
Summary
Exit Ticket:
Factor: 1
4x2 -
1
9
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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 – y
2
8. 64x2 – 25b
2
9. x2 – 100
10. x2 – 225
11. x4 – 64
12. x2 – 169
13. 16x2 – 81
14. x6 – 81
15. x2 – 49
.
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Factor by Grouping. 15. 16. 17. 18.
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Day 4: Factoring Completely
SWBAT: Factor a Trinomial Completely
Warm – Up Factor each. 1. 2.
3. 4. 5. Factor by Grouping.
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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.
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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
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Day 4 – Factoring Trinomials Completely Homework
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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.
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Example 1: Factoring Completely
Practice: Factoring Completely
FACTOR: 10x2 - 40
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Example 2: Factoring Completely
Factor: 8 Factor: 2
Practice: Factoring Completely 4. 10 5. 2 6.
4. x3 - 8x2 + 16x3. 4x2 + 24x + 36
7. 8.
9. 10.
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Challenge Problem:
Summary:
Exit Ticket:
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Day 5: Homework
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“REVIEW FOR TEST” SWBAT: Apply their knowledge on Factoring
Station # 1
Common Monomial Factors (GCF) Factor. 1) 9x2 – 21x5 2) 4x3 – 6x2 + 10x 3) Factor by Grouping. 4) 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
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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
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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 - 12p
2 + 35p. Find the factors that would
represent the length, width, and height of the rectangular prism.
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Chapter 8 Review
SWBAT: Apply Their Knowledge on Factoring.
A) 24t A) 2y4 A) 5n
9
B) 3t6 B) 2y
2 B) 3n
4
C) t2 C) y
3 C) 15n
D) t6 D) 2y D) 3n
9
4.
5. 6.
7. 8.
Factor each expression using the GCF.
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9. 10. 11.
A) (x + 6)(x + 1) A) (x - 3)(x - 7) A) (x + 5)(x + 10)
B) (x + 5)(x + 1) B) (x - 3)(x + 7) B) (x – 5)(x – 3)
C) (x - 5)(x + 1) C) (x + 10)(x + 11) C) (x + 5)(x + 3)
D) (x + 2)(x + 3) D) (x + 3)(x - 7) D) (x – 5)( x + 3)
Factor each binomial.
12. 13. 25x2 – 4 14.
A) (b - 8)(b - 2) A) (5x + 2)(5x - 2)
B) (b + 4)(b + 4) B) (15x + 2)(10x - 2)
C) (b + 8)(b + 2) C) (x + 2)(5x - 2)
D) (b - 4)(b + 4) D) (5x + 2)(5x + 2)
15. 16.
A) 3x3(x
2 - 9)
B) 3x3(x + 3)(x - 3)
C) 3x3(x + 3)(x + 3)
D) 9x3(x
2 - 9)
17.
36
18.
19.
20.
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21.
22. A box has a volume given by the trinomial + 3 – . What are the possible dimensions of the box?
Use factoring completely.
a. – – – – –
b. –
c. –
d. – –
