Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
Algebra 1 Unit 9 Notes: Polynomials and Factoring
1
Name_____________________________ Period _______
Unit 9 Calendar: Polynomials and Factoring
Students should be prepared for daily quizzes.
Each student is expected to do every assignment for the entire unit.
Students with no missing assignments at the end of the semester will be rewarded with
a 2% grade increase.
Students with no late or missing assignments will also get a pizza lunch at the end of
the semester.
HW reminders:
If you cannot solve a problem, get help before the assignment is due.
Extra Help? Visit www.mathguy.us or www.khanacademy.com.
Do you need to see the teacher notes? Do you need a copy
of a worksheet? Go to www.washoeschools.net/DRHSmath
for these items.
Day Date Assignment (Due the next class meeting)
Monday
Wednesday
2/26/18 (A)
2/28/18 (B)
9.1 Worksheet
Adding, Subtracting Polynomials, Multiplying
by a Monomial
Thursday
Friday
3/01/18 (A)
3/02/18 (B)
9.2 Worksheet
Multiplying Polynomials
Monday
Tuesday
3/05/18 (A)
3/06/18 (B)
9.3 Worksheet
Factoring by GCF
Wednesday
Thursday
3/07/18 (A)
3/08/18 (B)
9.4 Worksheet
Intro to Factoring Trinomials and Binomials
Friday
Monday
3/09/18 (A)
3/12/18 (B)
9.5 Worksheet
More Factoring Trinomials and Binomials
Tuesday
Wednesday
3/13/18 (A)
3/14/18 (B)
9.6 Worksheet
Factoring Completely
Thursday
Friday
3/15/18 (A)
3/16/18 (B) Unit 9 Practice Test
Monday
Tuesday
3/19/18 (A)
3/20/18 (B) Unit 9 Test
Algebra 1 Unit 9 Notes: Polynomials and Factoring
2
9.1 Notes: Adding, Subtracting Polynomials, Multiplying by a Monomial
Monomial
Binomial
Trinomial Polynomial
Note: All the exponents must be whole (positive) numbers!
Degree of a polynomial
Leading Coefficient
Descending order
Algebra 1 Unit 9 Notes: Polynomials and Factoring
3
Adding polynomials
Example 1: (4π₯3 + π₯2 β 5) + (7π₯ + π₯3 β 3π₯2)
Example 2: Find the sum: (π₯2 + π₯ + 8) + (π₯2 β π₯ β 1)
Subtracting polynomials
Example 3: Find the difference: (4π§2 β 3) β (β2π§2 + 5π§ β 1)
Example 4: Find the difference of (3π₯2 + 6π₯ β 4) β (π₯2 β π₯ β 7)
Example 5: You try! Simplify the expression: (3x2 + 5) β (x2 + 2) + (β 3m + 1)
Remember to
multiply each
term in the
polynomial by β 1
when you write
the subtraction
as addition.
Algebra 1 Unit 9 Notes: Polynomials and Factoring
4
Multiplying polynomials by monomials
Example 6: Find the product 3π₯3(2π₯3 β π₯2 β 7π₯ β 3)
Example 7: Multiply: βπ₯2(π₯ β 6) Example 8: Simplify: 3π¦3(π¦ β 4)
Example 9: An online store purchases boxes to ship their products. The large box has a volume of 4π₯3 +π₯2 + 5 units. The medium box has a volume of 2π₯3 + 3π₯ β 4 units. The store purchases one large box and
two medium boxes. What polynomial expression represents the total volume of the purchased boxes?
Algebra 1 Unit 9 Notes: Polynomials and Factoring
5
Example 10: Angela, Christie, and Mark each did the problem below. Who did the problem correctly, if
anyone? Describe the mistake made, if any, by each student.
Student work Describe the mistake, if any.
Angela (5π₯2 β 3π₯ + 7) β (3π₯ β 4π₯2) + (2π₯2 + 9 β 5π₯3)
= 5π₯2 β 3π₯ + 7 β 3π₯ β 4π₯2 + 2π₯2 + 9 β 5π₯3
=β5π₯3 + 3π₯2 β 6π₯ + 16
Christie (5π₯2 β 3π₯ + 7) β (3π₯ β 4π₯2) + (2π₯2 + 9 β 5π₯3)
= 5π₯2 β 3π₯ + 7 β 3π₯ + 4π₯2 + 2π₯2 + 9 β 5π₯3
=11π₯4 β 5π₯3 β 6π₯2 + 16
Mark (5π₯2 β 3π₯ + 7) β (3π₯ β 4π₯2) + (2π₯2 + 9 β 5π₯3)
= 5π₯2 β 3π₯ + 7 β 3π₯ + 4π₯2 + 2π₯2 + 9 β 5π₯3
=β5π₯3 + 11π₯2 β 6π₯ + 16
Example 11: Which of the following expressions is equivalent to 1
2π¦2(6π₯ + 2π¦ + 12π₯ β 2π¦)?
A. 9π₯π¦2
B. 18π₯π¦
C. 3π₯π¦2 + 6π₯
D. 9π₯π¦2 β 2π¦3
E. 3π₯π¦2 + 12π₯ β π¦3 β 2π¦
Example 12: Find h(x) = f(x) + g(x) if π(π₯) = (7π₯2 β 3π₯ + 2) and π(π₯) = (5π₯ β 2)
Example 13: Find h(x) = f(x)β g(x) if π(π₯) = (β2π₯3 β 4π₯ + 2) and π(π₯) = (5π₯3 + 5π₯2 β 2π₯)
Algebra 1 Unit 9 Notes: Polynomials and Factoring
6
9.2: Multiplying Polynomials
Warm-Up: Simplify each expression.
1) β3π₯3(2π₯2 β 9π₯) 2) (2π 3 β π 2 + 1) β (3π 2 β π + 4)
Multiplying
binomials
1) Distribute each term in the first binomial into the in the second binomial.
2) Combine like terms.
Example 1: Multiply the binomials (continued on the next page).
a) (π + π)(π + π)
b) (π + π)(π β π)
Algebra 1 Unit 9 Notes: Polynomials and Factoring
7
c) Multiply: (ππ + π)(π β π)
d) Multiply: (ππ β π)(π β ππ)
Example 2: Find h(x) = f(x) β g(x) if π(π₯) = (2π₯ + 7) and π(π₯) = (π₯ β 9).
Example 3: Multiply each expression.
a) (π₯ + 4)2 b) (π₯ β 7)2
c) (3π₯ + 4)2 d) (5π₯ β 1)2
Algebra 1 Unit 9 Notes: Polynomials and Factoring
8
Example 4: Simplify the expression: (π + 7)(π β 3) + (π β 4)(π + 5)
Example 5: Find each product.
a) (π₯ β 5)(π₯ + 5) b) (π¦ β 3)(π¦ + 3) c) (2π β 7)(2π + 7)
What do you notice about the products for Example 5?
Conjugates:
What happens when you multiply two conjugates?
Example 6: Write two binomial expressions which are conjugates and whose product equals π₯2 β 4.
Algebra 1 Unit 9 Notes: Polynomials and Factoring
9
Example 7: Multiply the polynomials.
a) (2πβ 5)(π2β 6πβ 3) b) (5p - 2)(3p2 β 2π + 1)
Example 8: You try! Find each product.
a) (3π₯ β 2)(4π₯2 β 5π₯ + 1) b) (2π2 + 3π β 2)(π β 4)
Example 9: Find h(x) = f(x) β g(x) if π(π₯) = (π₯2 β 4π₯ + 7) and π(π₯) = (3π₯ β 2)
Example 10: Simplify: 2(β4π + 9)2 + 5
Algebra 1 Unit 9 Notes: Polynomials and Factoring
10
Example 11: If π(π₯) = π(π₯) and π(π₯) = β3(π₯ + 1)2 + 4, then what polynomial represents π(π₯)?
Example 12: Which option below has a product of π₯2 β 4π₯ + 4? Choose all that apply.
A) (π₯ + 2)(π₯ β 2)
B) (π₯ β 2)(π₯ β 2)
C) (π₯ β 2)2
D) 2(π₯2 β π₯)
Example 13: What would you have to multiple (π₯ β 3) by to have a product of π₯2 β 9?
A) (π₯ β 3)
B) (π₯ β 9)
C) (π₯2 β 3)
D) (π₯ + 3)
Example 14: Which of the following expressions are equivalent to βπ₯2 + 3π₯ + 28? Select all that apply.
A) (π₯ + 4)(π₯ β 7)
B) β(π₯ + 4)(π₯ β 7)
C) (π₯ + 4)(7 β π₯)
D) (βπ₯ β 4)(π₯ β 7)
E) (βπ₯ β 4)(7 β π₯)
Algebra 1 Unit 9 Notes: Polynomials and Factoring
11
9.3: Factoring Out the Greatest Common Factor (GCF)
Exploration: What are the factors of each number below?
6 15 18
What is the greatest common factor of all three numbers?
What is the greatest common factor for each set of expressions below?
β8π₯ β20π₯3 β10π₯2
The GCF (Greatest Common Factor) is the largest common factor of two or more terms.
Factoring an expression by using the GCF:
Algebra 1 Unit 9 Notes: Polynomials and Factoring
12
Examples 1 β 6: Factor each expression by taking out the GCF.
1) 5x + 20 2) 8x β 4x2
3) 16π₯2π¦ + 40π₯y + 8π₯π¦2 4) β6π2 β 30π3
5) β12ππ + 32π 6) β8ππ₯3 + ππ₯2 β 3ππ₯
Examples 7 β 10: Factor each expression by taking out the GCF.
7) β4ππ β 2π2 8) β5π€π₯3 + 10π€π₯2
9) 6π¦ β 15π¦3 10) β9ππ3 + ππ2 β 2π4π
11) One factor of β7π₯3π¦ β 21π₯2π¦2 is (β7π₯2π¦). What is the other factor?
Algebra 1 Unit 9 Notes: Polynomials and Factoring
13
12) Factor: 15π₯3 β 7π¦4 β 2π§
For #13 β 15: Find each product. Try to do this without showing any work!!
13) (π₯ + 2)(π₯ + 3) 14) (π¦ + 4)(π¦ + 7) 15) (β β 3)(β + 5)
Challenge! What are the factors of each trinomial? (Try to work backwards to figure this out!)
16) π₯2 + 6π₯ + 8 17) π₯2 + 7π₯ + 10
Example 18: Write a polynomial expression to represent the area of the rectangle shown below, if π΄ = πβ.
Example 19: A rectangle has an area that can be represented by (3π₯3 + 9π₯2 + 6π₯) ππ‘2. If the height of the
rectangle is 3π₯ ππ‘, then what expression can represent the base?
Algebra 1 Unit 9 Notes: Polynomials and Factoring
14
Example 20: Given the triangle shown to the right, write a
polynomial expression to represent the perimeter of the triangle.
9.4 Notes: Intro to Factoring Trinomials and Binomials Warm-Up. Simplify the following:
1) (π₯ β 3)(π₯ + 3) 2) (x + 3)(x β 4)
Algebra 1 Unit 9 Notes: Polynomials and Factoring
15
Work in groups to multiply (expand) the following expressions:
(π₯ + 5)(π₯ β 3) (π₯ + 2)(π₯ + 8) (π₯ + 4)(π₯ β 4)
Factoring a trinomial is the inverse (opposite) of multiplying binomials.
Example 1: Factor x2 + 10x + 16 Check by multiplying your answer:
Examples 2 β 4: Factor each expression.
2) x2 + 10x + 9 3) a2 β 6a + 9 4) π₯2 + 7π₯ β 30
You try! Examples 5 β 7: Factor each expression.
5) π₯2 + 10π₯ + 9 6) π¦2 β π¦ β 6 7) π₯2 + 2π₯ + 1
Algebra 1 Unit 9 Notes: Polynomials and Factoring
16
Some expressions have a GCF that need to be factored out BEFORE you factor the trinomial.
Example 8: Factor the expression below completely.
4y2 + 12y β 40
Step 1: Factor out the GCF:
Step 2: Factor the remaining trinomial.
(Make sure to leave the GCF as part of your answer.)
Examples 9 β 12: Factor each expression completely.
9) βπ₯2 + 4π₯ + 12 10) π€3 β 10π€2 + 25π€
11) β2π₯2 + 14π₯ β 24 12) 2π2π β 10ππ + 8π
Do you remember how to multiply conjugates? Multiply each expression below. (Try to do this
without work!)
(π₯ β 5)(π₯ + 5) (π₯ + 11)(π₯ β 11)
Factoring Difference of Two Perfect Squares:
Algebra 1 Unit 9 Notes: Polynomials and Factoring
17
For #13 β 16: Factor each expression.
13) π₯2 β 25 14) π2 β 49π2 15) 36 β π¦2 16) π₯6 β 100
You try! For #17 β 20: Factor each expression.
17) π2 β 4 18) 1 β π2 19) π2 β 81π2 20) π10 β 9
Sometimes we need to factor out the GCF before we factor the difference of two perfect squares.
Also, not all expressions factor. If an expression does not factor at all, then it is _______________.
Examples 21 β 29: Factor each expression completely.
21) 5π₯2 β 20 22) β2π₯3 + 32π₯ 23) 4π5 β 4π3
24) π2 + 16 25) β6π4 + 36 26) βπ₯5 + 9π₯3
27) 3π₯2 β 24π₯ + 12 28) βπ₯3 + 6π₯2 + 16π₯ 29) π5 + 3π4
Algebra 1 Unit 9 Notes: Polynomials and Factoring
18
9.5: More Factoring Trinomials and Binomials
Warm-Up: Simplify each expression. Try to do these without showing work!
π) (ππ β π)(π + π) 2) (ππ + π)(ππ β π)
Factoring Trinomials with a leading coefficient different than one:
Example 1: Factor 2π₯2 β 11π₯ + 5 Check your solution by using multiplication:
Algebra 1 Unit 9 Notes: Polynomials and Factoring
19
Examples 2 β 3: Factor each expression.
2) 3n2 + 2n - 8 3) 2y2 β 13y β 7
Examples 4 β 6: Factor each expression.
4) 9π¦2 + 6π¦ + 1 5) 6π₯2 + 5π₯π¦ β 6π¦2 6) 15π₯2 β π₯ β 6
You Try! Examples 7 β 9: Factor each expression.
7) 3π₯2 β 5π₯ + 2 8) 8π₯2 + 14π₯ β 15 9) 2π2 + ππ β 21π2
Factoring Binomials with a leading coefficient different than one:
Examples 10 β 12: Factor each expression.
10) 25π₯2 β 4 11) 49π4 β 9π2 12) 36π2 β π6
Algebra 1 Unit 9 Notes: Polynomials and Factoring
20
You try! For examples 13 β 15, factor each expression.
13) 121β2 β 4π8 14) 25 β 16π2 15) 169π₯2 β 49π¦12
If you are able to factor out a GCF from an expression, always do that first!
Examples 16 β 18: Factor each expression completely. (Hint: look for a GCF first!)
16) 6π₯2 β 2π₯ β 4 17) 36π5 β 9π3 18) β4π₯3 + 4π₯2π¦ + 3π₯π¦2
You Try! For examples 19 β 21, factor each expression completely.
19) 28π₯2 + 38π₯π€ β 6π€2 20) β6π₯2 + 12π₯ + 90 21) 25π₯4 β 100π₯2
Algebra 1 Unit 9 Notes: Polynomials and Factoring
21
9.6 Notes: Factoring Completely
Warm-Up
1) Factor: π₯2 + 5π₯ + 6 2) Factor: π₯2 β 64
3) Simplify: (π₯ + 7)2 4) Simplify: (4π₯2 β 3π₯ + 7) β (7π₯2 β 6π₯ + 2)
Algebra 1 Unit 9 Notes: Polynomials and Factoring
22
Examples 1 β 4: Factor each polynomial completely.
1) 5π4β 405 2) 2π₯2 β 8π₯ β 10
3) π₯4 β 16 4) βπ₯3 β π₯2 + 12π₯
Examples 5 β 10: Factor completely.
5) 3π3 β 21π2 + 30π 6) 81π5 β π 7) 2π₯2 + 5π₯π¦ + 2π¦2
Factoring
Completely
Step 1: If possible, factor out a Greatest Common Factor.
Step 2: Can you factor the binomial or trinomial any further?
Step 3: Keep factoring until each portion of your answer is fully factored.
Algebra 1 Unit 9 Notes: Polynomials and Factoring
23
8) 2y4 β 32 9) 49y2 β 25w6 10) βπ₯3 β 2π₯2 + 15π₯
11) Given (π₯ + 4) is a factor of 2π₯2 + 11π₯ + 2π, determine the value of m.
12) Which of the following expressions are equivalent to βπ₯2 + 4π₯ + 21? Select all that apply.
A) (π₯ + 3)(π₯ β 7)
B) β(π₯ + 3)(π₯ β 7)
C) (π₯ + 3)(7 β π₯)
D) (βπ₯ β 3)(π₯ β 7)
E) (βπ₯ β 3)(7 β π₯)
Algebra 1 Unit 9 Notes: Polynomials and Factoring
24
Work with your group to match each polynomial to its factors.
Polynomials
1. 8π₯2 β 63π₯ β 81
2. 49π₯2 β 25
3. 8π₯2 β 2π₯ β 45
4. 2π₯2 + 11π₯ + 12
5. π₯2 β 7π₯ β 8
6. π₯2 + 6π₯ β 27
7. π₯2 β 4
8. 7π₯2 + 32π₯ β 60
Factors
A. (π₯ + 9)
B. (7π₯ β 5)
C. (π₯ β 2)
D. (4π₯ + 9)
E. (8π₯ + 9)
F. (7π₯ + 5)
G. (π₯ β 9)
H. (2π₯ + 3)
I. (7π₯ β 10)
J. (π₯ + 6)
K. (π₯ β 3)
L. (2π₯ β 5)
M. (π₯ + 2)
N. (π₯ β 8)
O. (π₯ + 1)
P. (π₯ + 4)
Are you done? Get your answers checked off, and then complete the Factoring Card Match with a partner.