Upload
others
View
38
Download
4
Embed Size (px)
Citation preview
1
UNIT 9 (Chapter 7 BI) –Polynomials and Factoring Name:_____________________ The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay attention and make any necessary changes. All assignments are due the following class
period unless indicated otherwise. Monday Tuesday Wednesday Thursday Friday
March 16
Review
17
Unit 8 Test
18
Pretest
Section 7.1
19
Section 7.2
20
Section 7.3
23
Section 7.4
24
Section 7. 5a
25
Section 7.5b
26
Section 7.5c
Review
27
Quiz 7.1-7.5
End of 3rd MP
30
Section 7.6a
Box Method
31
Section 7.6b
April 1
Section 7.6c
2
Section 7.7
3
Break Begins
13
Section 7.8
14
Review
15
Review
16
Unit 9 Test
17
Section Page Assignment
7.1 p. 362 #6-52 even 55-57
7.2 p. 369 #4-44 even, 47,48
7.3 p. 375 #2-34 even, 37-42
Review --- Worksheet in packet
7.4 & 5a p. 381 #4-40 even, 49-52
7.5b p. 389 #12-23, 28-40 even, 52-58
7.5c --- Worksheet in packet
7.6a
7.6b --- Worksheet in the packet
7.6c --- Worksheet in packet
7.7 p. 401 #
7.8 p. 407 #
2
Lesson 7.1– Adding and Subtracting Polynomials Algebra 1
Essential Question How can you add and subtract polynomials?
Warm-Up Exercise
(a) 2452 xx (b) 15)4(3 xx (c) )8(9 x
Core Concepts
Polynomials
A polynomial is a monomial or a sum of monomials. Each monomial is called a term of the
polynomial. A polynomial with two terms is a binomial. A polynomial with three terms is a
trinomial.
Binomial Trinomial
5 2x 2 5 2x x
The degree of a polynomial is the greatest degree of its terms. A polynomial in one
variable is in standard form when the exponents of the terms decrease from left to
right. When you write a polynomial in standard form, the coefficient of the first term is
the leading coefficient.
1. Examining Polynomials
Decide whether or not it is a polynomial. If it is, then write it in standard form, list its degree,
leading coefficient, and constant term.
(a) (b) (c) 8 (d)
Classifying Polynomials Polynomials can be classified according to their degree and by the number of terms. Fill out the chart below.
Polynomial Degree Classified by Degree Classified by # of terms
A.
B.
C.
D.
E.
F.
3
2. Classifying Polynomials
Classify each polynomial by its degree and by its number of terms.
(a) x4
(b) 32 7xx
(c) 37 2 xx
3. Adding Polynomials
Add the polynomials using a vertical format in
part (a) and a horizontal format in part (b).
(a)
(b)
4. Subtracting Polynomials
Subtract the polynomials using a vertical format in
part (a) and a horizontal format in part (b).
(a)
(b)
4
Lesson 7.2 – Multiplying Polynomials Algebra 1
Essential Question How can you multiply two polynomials?
Warm-Up Exercise
Simplify the Expression.
(a) )6()3( xx (b) )6(3 xx (c) 107)312( x (d) )(332 xx
1. Investigating Binomial Multiplication
Use the diagram at the right and determine the area of the
entire region. Then complete the statement below.
____________ ____________ = __________________________
2. Multiplying Binomials Using the Distributive Property
Find the product using the distributive property.
(a) )3)(2( xx (b) )35)(2( 2 xxx
(c) )15)(32( xx (d) )52)(134( 2 xxx
x
x
x
1 1
1
5
3. Multiplying Binomials Using F.O.I.L. Pattern
Find the product of binomials using the F.O.I.L. method.
(a) )15)(32( xx
(b) )12)(43( xx (c) )45)(23( xx
In Exercises 4–9, use a table to find the product.
4. 3 2x x 5. 1 6y y 6. 3 7q q
7. 2 5 3w w 8. 6 2 3 2h h 9. 3 4 3 4j j
6
Lesson 7.3 – Special Product of Polynomials Algebra 1
Essential Question What are the patterns in the special products
( a + b)2 and (a + b)( a – b)?
Warm-Up Exercise
Simplify the Expression.
(a) 2)2
1( x (b) 23 )6( m (c) 22 )
5
2( y (d) 2)2( xx
1. Using Sum and Difference Binomial Patterns
Find the product.
(a) )53)(53( bb (b) )25)(25( dd
2. Special Product: Squaring Binomials
Find the product.
(a) 2)43( n (b) 2)72( yx
(c) 2)43
1( a
7
7.1-7.3 Review Worksheet Name _______________________
Identify the leading coefficient, constant term and classify the polynomial by degree and by
number of terms.
1. 5x – 2 2. 8x3 – 3x + 45 3. – 6x + 5x4 - 3x
Add or subtract .
4. ( 5x3- 3x) – (7x2 – 3x + 1) 5. ( 2x3+ 4) – (- x2+ 3x)
6. ( m + 3m3- 4m5) + (2m3 + 5m5 - 4) 7. (x2+ 1) + (-3x2 – 7) - ( x2 + 3x)
8. (5x2+ 4) - (3x + 7) + ( 2x2 -1) 9. 2(x2- 4x + 5) – ( x2 + 6x – 1)
Find the product.
10. -5x ( 2x – 5) 11. ( 4x2 – 2x)( 8)
8
12. ( 3x - 5)(4x + 1) 13. ( 2x – 1)( 3x3 – 2x + 7)
14. 4(x – 5)( 3x + 4) 15. ( 5x – ½ ) ( 4x - 3)
16. ( 3x3 + 1)( 4x - 7) 17. ( 2x3 + 3x2 – 1) (3x3 – 3x + 2)
Write the square of the binomial as a trinomial.
18. ( x + 5)2 19. ( 3y – 4)2
Find the product.
20. ( x + 7)(x – 7) 21. (3m – 5) (3m + 5)
22. ( ½ x + 4)( ½ x – 4)
9
Lesson 7.4 – The Zero Product Property Algebra 1
Essential Question How can you solve a polynomial equation?
Work with a partner. Substitute 1, 2, 3, 4, 5, and 6 for x in each equation and determine whether the
equation is true. Organize your results in the table. Write a conjecture describing what you discovered.
Equation x = 1 x = 2 x = 3 x = 4 x = 5 x = 6
a. 1 2 0x x
b. 2 3 0x x
c. 3 4 0x x
d. 4 5 0x x
e. 5 6 0x x
f. 6 1 0x x
Work with a partner. The numbers 0 and 1 have special properties that are shared by no other
numbers. For each of the following, decide whether the property is true for 0, 1, both, or neither.
a. When you add ____ to a number n, you get n.
b. If the product of two numbers is ____, then at least one of the numbers is 0.
c. The square of ____ is equal to itself.
d. When you multiply a number n by ____, you get n.
e. When you multiply a number by n by ____, you get 0.
f. The opposite of ____ is equal to itself.
One of the properties in Exploration 3 is called the Zero-
Product Property. It is one of the most important properties in
all of algebra. Which property is it? Why do you think it is
called the Zero-Product Property? Explain how it is used in
algebra and why it so important.
1 EXPLORATION: Writing a Conjecture
2 EXPLORATION: Special Properties of 0 and 1
10
1. Using the Zero Product Property
Use the Zero Product Property to solve the
equations written in factored form.
(a) 0)9)(4( xx
(b) 0)23)(12( xx
(c) 0)1)(5(43
21 xx (d) 0)8(2 xx
2. Zero Product Property with Special Products
Use the Zero Product Property to solve the equations written in factored form.
(a) 0)7( 2 x (b) 0)6( 2
41 x
(c) 0)10)(3)((543 xxx (d) 0)1)(11()53(
852 xxx
Zero Product Property
Factored Form
11
Lesson 7.5a– Factoring the Greatest Common Factor (GCF) Algebra 1
Warm-up Exercise – Use the Distributive Property.
(a) )964(5 2 xx (b) )487(2 2 xxx (c) )5(7 yxxy
In Exercises 3- 5, solve the equation.
3. 26 0k k 4. 235 49 0n n 5. 24 52 0z z
6. A boy kicks a ball in the air. The height y (in feet) above the ground of the ball is
modeled by the equation 216 80 ,y x x where x is the time (in seconds) since
the ball was kicked. Find the roots of the equation when y = 0 . Explain what the
roots mean in this situation.
Common Monomial Factoring
You can think of common monomial factoring as
“reversing” the distributive property. Our goal here
is to factor, or pull out, the greatest common
monomial factor. We call this factoring the GCF.
Ex.
1. Factor the GCF out of each polynomial.
(a)
(b)
2. Factor the GCF out of each polynomial.
(a) (b)
(c) (d)
12
7.5A Practice: Factoring the Greatest Common Factor Name:
In each of the polynomials, factor out the GCF. If there is no GCF, then just write “No GCF.”
1. xx 102 2 2. xx 189 4 3. xxx 4128 25
4. vv 186 3 5. qq 124 4 6. 293 xx
7. 35 624 tt 8. 235 284 aaa 9. ddd 3618 26
10. 10x5+24x3-16x2 11. 36x8+90x7+48x2 12. 18x8+10x6-10x5
13. 36x10-28x5-20x 14. 40x5-20x3+15x2 15. 50x8-45x3-30x2
16. 16x8+14x6-14x7 17. 30x6-40x4-15x8 18. 15x7+21x5-90x
13
Lesson 7.5b– Factoring x2 + bx + c Algebra 1
Essential Question How can you use algebra tiles to factor the trinomial
2x bx c into the product of two binomials?
Factoring a Quadratic Trinomial (x2 + bx + c)
To factor a quadratic expression means to write it as the
product of 2 linear expressions. You can think of factoring
trinomials of the form x2 + bx + c as “reverse FOILing.”
(x + p)(x + q) =
(x + p)(x + q) =
In order to factor x2 + bx + c, you must find p and q
such that: (p + q) = ___ AND pq = ___
Example: x2 + 6x + 8 = (x + ? )(x + ? )
1. Factor the following
(a) x2 + 3x + 2
(b) x2 - 8x + 15
(c) x2 + 11x + 10
(d) x2 - 8x – 9
(e) x2 + 3x – 18
(f) x2 - 17x + 60
(g) y2 – 2y - 48
Solving Quadratic Equations by
Factoring
Example: x2 – 3x = 10
a. Rewrite
b. Factor
c. Use
d. Solve
14
2. Solve the Equation by Factoring
(a) z2 + 11z = 26 (b) x2 + 11x +18 = 0 (c) x2 + 16x = -15
3. Factoring Trinomials with a G.C.F.
Completely factor the trinomial.
(a) 45243 2 xx (b) xxx 24306 23
(c) 345 11012111 xxx (d) yxyxyx 8910 56147
4. Solving Equations by Factoring Trinomials and a G.C.F.
Solve each equation using the Zero Product Property.
(a) 090639 2 xx (b) 090213 23 yyy
(c) 4234 2963 xxxx (d) 147707 2 yy
5. The area of a right triangle is 32 square miles. One leg of the triangle is 4 miles longer than
the other leg. Find the length of each leg.
15
Name ________________________________________Date _____________
7.5 Worksheet – Factoring Common Monomials, Factoring Trinomials, and Solving Equations
Factor the GCF for each polynomial.
1. xx 255 2 2. 57 1512 xx 3. xxx 4124 26
4. vv 1821 3 5. 39 336 qq 6. 63620 xx
Factor each trinomial of the form cbxx 2
7. 862 xx 8. 432 xx 9. 232 xx
10. 822 xx 11. 1272 xx 12. 562 xx
13. 202 xx 14. 1682 xx 15. 24102 xx
Factor the GCF first, and then factor the remaining trinomial.
16. 234 6xxx 17. 345 158 xxx 18. xxx 30162 23
16
19. xxx 16204 23 20. 234 8422 xxx 21. 345 64244 xxx
22. 345 128322 xxx 23. xxx 108393 23 24. 234 108453 xxx
Solve the following equations by factoring.
25. 04032 xx 26. 063162 xx 27. xx 672
28. 052 23 xx 29. 018122 23 xxx 30. 2234 1224123 xxxx
17
Lesson 7.6(a) – Factoring Quadratic Trinomials:
ax2 bx c Algebra 1
Essential Question How can you use algebra tiles to factor the trinomial2 + +ax bx c into the product of two binomials?
Warm-Up Exercise
Factor each expression completely.
(a)
x2 6x 40 (b)
2x2 12x 10
Find each product using the distributive property.
(c)
2x 3 3x 1 (d)
x 5 4x 7
Example:
3x2 2x 8
Step 3:
Step 4:
Factoring: The Box Method- Steps
1. Insert the first term of the trinomial into the upper left box.
2. Insert the last term into the lower right box.
3. Find the product of the leading coefficient and the constant term.
Be sure to carry all negative signs (if necessary)
4. Find and list all factors of the product from step 3.
5. Find the pair of factors that sum to the middle term’s coefficient.
6. Insert each factor into the empty boxes as x terms.
7. Find the GCF of each row and each column
- If the front box of each row or column is negative, then the GCF is negative
- If there is nothing in common, then the GCF is 1
8. Write the GCF’s from the rows as a binomial, and write the GCF’s form the
columns as a binomial.
Factoring: The Box Method
When the leading coefficient of a trinomial is greater than 1, the factors of both the
leading coefficient and the constant play a role in determining. The box method
helps us organize the work needed to factor each expression.
This method will only work if the greatest common factor is factored out first. Step 1
Step 2
Step 5
Step 7
18
1. Factoring Using the Box Method
Factor each expression completely.
(a)
9x2 15x 4 (b)
3x2 10x 8 (c)
12y2 17y 6
Practice Problems - Complete all problems in the space below. Show all work
Factor each expression completely.
(a)
3b2 11b 6 (b) 584 2 xx
(c) )3108(5 2 mm
19
Worksheet 7.6(a) – Factoring Quadratic Trinomials:
ax2 bx c Name_________________
Factor each polynomial of the form cbxax 2 . Use the “Box” method to factor.
1. 2116 2 xx 2. 295 2 xx 3. 10116 2 xx
4. 44373 2 yy 5. 41514 2 yy 6. 706 2 xx
7. 10192 2 xx 8. 12317 2 xx 9. 2032 2 xx
10. 4113 2 xx 11. 8103 2 xx 12. 672 2 xx
13. 352 2 xx 14. 130 2 xx 15. 592 2 xx
20
Lesson 7.6(b) – Solving Quadratic Equations w/ Zero Product Property Algebra 1
Warm-up Exercises
Factor the following expressions
(a) 3x2 – 4x – 7 (b) –11 + 2x2 + 21x (c) 21x –11 + 2x2
1. Solving Quadratic Equations Using Zero Product Property
Solve the equation
(a) – 21x + 4x2 + 5 = 0
(b) 7 + 14n + 21n2 = 6n + 11 (c) – 10t + 5t2 = -11t2 – 2t + 48
(d) – x + x2 – 8 = 82
1. Factoring Polynomials Involving the Greatest Common
Factor
Factor each expression completely.
(a)
14x2 32x 8 (b)
72n2 6x 45 (c)
12x4 26x3 10x2
Solving Quadratic Equations 1.
2.
3.
21
2. Completely Factoring Polynomials of Higher Degrees
Factor each expression completely.
(a) 6113 2 bb (b) 584 2 xx
(c) 155040 2 mm (d) xxx 279648 23
7.6C Worksheet: Factoring the GCF and cbxax 2 Name:
In each of the polynomials, factor out the GCF and then factor the remaining
trinomial.
1. 62712 2 xx 2. 345 8103 xxx 3. xxx 2010840 23
4. 21012 2 xx 5. 256020 2 xx 6. 23 412 ddd
22
7. 324 693615 ggg 8. 456 15148 xxx 9. 5018090 2 xx
10. 48816 2 xx 11. 234 32130 xxx 12. xxx 202040 23
13. 243624 2 xx 14. 345 2032 xxx 15. 1293 2 xx
23
Solve the equations by factoring.
16. 6x2 + 13x + 5 = 0 17. 3x2 + 7x = -2
18. 10x2 = 5 = - 15x 19. 12x2 + 32x = -5
20. 6x2 – 10x – 4 = 0 21. 6x2 – 27x + 27 = 0
22. 8x2 + 10x + 3 = 0 23. 4x2 – 8x – 5 = 0
24
Lesson 7.7 – Difference of Squares and Perfect Square Trinomials Algebra 1
Essential Question How can you recognize and factor special products?
Warm-Up Exercise
Find the product.
(a) )12)(12( xx (b) )12)(12( yy (c) 2)13( n (d) 2)32( n
1. Factoring the Difference of Two Squares
Factor each expression.
(a) 642 x (b) 94 2 x
(c) 236121 y (d) 16250 2 a
2. Factoring Perfect Squares Trinomial
Factor each expression and identify the pattern for each perfect square trinomial.
(a) 962 xx (b) 24246 2 xx
(c) 100020010 2 yy (d) 92416 2 yy
3. Solving Equations Involving Special Product Factorization
Solve the equation.
(a) 025 2 x (b) 221812 xx
Difference of Two Squares
Since the special product of two binomials has the
following property:
The difference of two squares reverses the process by
factoring –
________________
Perfect Square Trinomial
Since the special product of two binomials has the
following property:
By reversing the process –
________________
________________
25
Lesson 7.8 – Factoring a Polynomial Completely Algebra 1
Essential Question How can you factor a polynomial completely?
Work with a partner. Match the standard form of the polynomial with the equivalent factored form on
the next page. Explain your strategy.
a. 3 2x x b. 3x x c. 3 2 2x x x
d. 3 24 4x x x e. 3 22 3x x x f. 3 22x x x
g. 3 4x x h. 3 22x x i. 3 2x x
j. 3 23 2x x x k. 3 22 3x x x l. 3 24 3x x x
m. 3 22x x n. 3 24 4x x x o. 3 22x x x
Factoring a Polynomial Completely
A polynomial is completely factored if it can be written as
the product of monomials and prime polynomials.
Ex.
Prime Polynomials
1 EXPLORATION: Matching Standard and Factored Forms
A. B. C.
D. E. F.
G. H. I.
J. K. L.
M. N. O.
26
1. Factor the Polynomials Completely.
If it is prime, then say so.
(a) 25 12133 xx (b) 94 2 x (c) xxx 30255 23 (d) 24 375 xx
(e) 132 xx
2. Factoring By Grouping
Factor the polynomials
(a) 2464 23 xxx (b) 72362 23 xxx
3. Solving Equations By Factoring Completely
Find all solutions to each equation.
(a) 010016 3 xx (b) 0252523 xxx
(c ) 23 21 30 0x x (d) 25 5 30 0y y ( e ) 4 281 0c c
Factoring By Grouping You can use the distributive property to factor
some polynomials that have FOUR terms.
27
Unit 9 Test Review
I. Identifying Polynomials
Name each polynomial by its degree, number of terms, and leading coefficient
1. 5x – 2 2. 336 x 3. 725 432 xxxx
II. Polynomial Operations
Perform the indicated operation for each polynomial expression below.
4. 13735 23 xxxx 5. 541 22 xx 6. 5353 5243 mmmmm
7. 47735 222 xxx 8. 753283 22 tttt 9. 13125 44 xxxx
10. 3625 2 xxx 11. 482
14 32
xx 12. 4725 xx
13. 5143 2 xxx 14.
2
12864 2 xxx 15. 5252 xx
16. 274 x 17. xxxx 4343 22 18. 2
115 x
28
Guidelines for Factoring Polynomials Completely
To factor a polynomial completely, you should try each of these steps.
1. Factor out the greatest common monomial factor. 23 6 3 2x x x x
___________________________________________________________________
2. Look for a difference of two squares or a perfect
square trinomial. 22 4 4 2x x x
___________________________________________________________________
3. Factor a trinomial of the form 2 ax bx c into a product
of binomial factors. 23 5 2 3 1 2x x x x
___________________________________________________________________
4. Factor a polynomial with four terms by grouping.
3 2 24 4 1 4x x x x x
___________________________________________________________________
I. Factoring Polynomials.
Factor each expression completely
1. 35 1824 tt 2. 34 213 ww 3. 1012 644 xx
4. 457 3312111 xxx 5. 6910 121212 yyy 6. 7212 x
7. 72172 xx 8. 100202 xx 9. 72382 xx
10. 21544 xx 11. xx 902 12. xxx 23010
29
13. yyy 12102 23 14. 234 6022 ddd 15. mmm 21287 23
16. 212 2 xx 17. 169 2 xx 18. 734 2 xx
19. 144921 2 xx 20. 106424 2 xx 21. xxx 162024 23
22. 259 2 x 23. 964 2 x 24. 3162200 xx
25. 162362 2 xx 26. 2164836 xx 27. 100609 2 xx
30
28. 4423 xxx 29. 1234 23 xxx 30. 153102 23 ddd
Solve each equation. Be sure to find all solutions.
31. 064122 xx 32. 2832 xx 33. 014322 xx
34. 010113 2 xx 35. 81012 2 xx 36. xx 1356 2
37. 0483 3 xx 38. xx 20254 2 39. 01892 23 ddd