Unit 8 – Polynomials 1 Name: ____________________ Teacher: _____________ Per: ___ Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10 – Unit 8 – [Polynomials]
Unit 8 – Polynomials 1
Name: ____________________ Teacher: _____________ Per: ___
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9
Unit 10
– Unit 8 – [Polynomials]
Unit 8 – Polynomials 2
To be a Successful Algebra class,
TIGERs will show…
#TENACITY during our practice, have…
I attempt all practice I attempt all homework I never give up when I don’t understand
#INTEGRITY as we help others with their work, maintain a…
I always check my answers I correct my work, I never just copy answers I explain answers, I never just give them
#GO-FOR-IT attitude, continually…
I write down all notes, even if I’m confused I remain positive about my goals I treat each day as a chance to reset
#ENCOURAGE each other to succeed as a team, and always…
I offer help when I understand the material I push my teammates to reach their goals I never let my teammates give up
#REACH-OUT and ask for help when we need it!
I ask my questions during homework check I ask my teammates for help during practice I attend enrichment/tutorials when I need to
Unit 8 – Polynomials 3
Unit Calendar
MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY
February 9 10 11 12 13
…
…
Polynomial Vocabulary
Add and Subtract Polynomials
Multiplying Polynomials
16 17 18 19 20
Holiday
Multiplying Polynomials
QUIZ
Multiplying Polynomials
Early Release
23 24 25 26 27
Factor GCF Factor GCF Factor
Factor
QUIZ
March 2 3 4 5 6
Factor
Factor
Review
Review
TEST
Essential Question
How can I use what I know about operations with real numbers to help me with
operations with algebraic expressions?
Unit 8 – Polynomials 4
Critical Vocabulary
Polynomial
Term
Degree
Factor
Unit 8 – Polynomials 5
Polynomials: Basics and Vocabulary
Term
A combination of _______________, ______________ and ______________.
Like Term
Terms that have the same ________________ and ______________ combination.
Polynomial
A combination of one or more terms named by their ________ and number of _______.
Degree Name:
Degree Example Name
0
1
2
3
4 or more
Number of Terms Name:
Terms Example Name
1
2
3
4 or more
Unit 8 – Polynomials 6
Examples:
Classify each polynomial by its degree and nuber of terms:
26 5 2x x
15 34 2g g
Combine like terms and arrange in descending order:
2𝑥2 + 5𝑥 + 6 − 3𝑥2
2 2 2a 2b 4a 2 8a 2b 7a 4a 2
Practice:
Classify each polynomial by its degree and nuber of terms:
6c
22 5 1a a 29 5x x
6
2 4 32x x x x 3x
3 25 10 4n n n
3 23 3 4x x x 210 1x
Combine like terms and arrange in descending order:
24x 1 3x 5 2x
3xy 5 2xy 10 −4𝑥 + 10𝑥
7𝑥2𝑦 + 3𝑥2 − 2𝑦2 + 7𝑥2 − 5𝑥2𝑦
12𝑟 + 5 + 3𝑟 − 5 𝑥2 + 4𝑥 − 3𝑥
5𝑥 + 3𝑥 + 7 − 𝑥 − 4
𝑥2 − 3𝑥 + 4 + 3x – 2 𝑥2 + 5𝑥 − 3 − 6𝑥 − 2𝑥2
Unit 8 – Polynomials 7
Unit 8 – Polynomials 8
Polynomials: Adding and Subtracting
Use algebra tiles to find the sum. (𝑥2 − 𝑥 + 1) + (6𝑥 − 3) (3𝑥2 + 2𝑥 + 5) + (−𝑥2 − 𝑥 − 4)
Examples:
(6𝑥2 + 3𝑥) + (2𝑥2 + 6𝑥)
(−6𝑥3 + 5𝑥) + (4𝑥3 + 𝑥2 − 2𝑥 + 9)
(𝑎𝑏2 + 13𝑏 − 4𝑎) + (3𝑎𝑏2 + 𝑎2𝑏 + 𝑎 + 7𝑏
(11ℎ𝑧3 + 8ℎ𝑧) − (9ℎ𝑧3 − 3ℎ𝑧 − 𝑧)
Find the perimeter of a rectangle with length 4x + 3 and width 2x – 1.
The perimeter of a triangle is 6x2 + 5x + 9. If two of the sides are 3x2 + x + 2 and x2 + 2x + 1, find the missing side.
+ +
Unit 8 – Polynomials 9
Practice:
(2𝑦2 + 3𝑦) + (𝑦2 + 7𝑦)
(9𝑥3 − 5𝑥) − (3𝑥)
(2𝑥3 + 4𝑥 − 2) − (4𝑥3 − 6)
(−5𝑦2 + 7𝑦 − 2) + (4𝑦2 + 𝑦 + 8)
(4𝑥3 + 𝑥2) + (2𝑥3 − 3)
(𝑡3 − 2𝑡) − (𝑡2 − 2𝑡 + 6)
(12𝑑2 + 3𝑑𝑥 + 𝑥) − (−4𝑑2 + 2𝑑𝑥 − 8𝑥)
(4𝑥3𝑦 − 𝑥2 + 4𝑥) + (𝑥3𝑦 − 𝑥2 − 4𝑥)
Find the perimeter of a triangle with side lengths 2x2 + 3x – 1, 5x + 2, and 3x2 + 5.
The perimeter of a rectangle is 10x + 6. If the width is 2x + 1, what is the length? 2x + 1 2x + 1
Unit 8 – Polynomials 10
Unit 8 – Polynomials 11
Polynomials: Multiplying
Use algebra tiles to find the product.
4(𝑥 − 3) (2𝑥)(3𝑥) (𝑥 + 1)(𝑥 − 2)
Examples:
5𝑥(𝑥 − 3)
−4𝑥(𝑥2 + 8) (𝑥 + 4)(𝑥 − 5) (3𝑛 + 2)(6𝑛 + 1)
Unit 8 – Polynomials 12
Practice:
3𝑥(𝑥 + 8)
2𝑥(−2𝑥4 − 3) 8𝑦(𝑥 − 3𝑦)
(𝑥 − 2)(𝑥 + 8)
(𝑥 − 3)(𝑥 − 6) (𝑥 − 7)(𝑥 + 7)
(2𝑥 + 1)(𝑥 + 2)
(𝑥 + 3)(𝑥 − 4) (3𝑥 − 4)(2𝑥 − 5)
Unit 7 Multiplication Practice:
(3𝑥)(𝑥2𝑦3)
(2𝑎𝑏)(5𝑎3𝑏)
Unit 8 – Polynomials 13
Polynomials: Multiplying Cont…
Examples:
( 5b – 3 )( 2b + 1 )
(3 – 4x)(8 + 3x)
(-2x + 7y)(3x – 5y)
(2x – 5)(4x2 - 3x + 1)
Unit 8 – Polynomials 14
Practice:
(2x – 1)(x + 3)
( 4n + 3 )( 3n - 2 ) (x – 3)(x2+3x+2)
(2 + 5z)(3 – 4z)
(3m + 2)(5m – 6) ( x - 7)( x² + 7x – 4 )
(-6a +5b)(5a – 4b)
(2y - 3)(6y – 7) (3x + 2)(2x2 - 5x - 7)
Unit 8 – Polynomials 15
Polynomials: Multiplying Cont…
Examples:
(5𝑥 + 2)(5𝑥 − 2)
(4𝑥 − 3)2
(𝑦 − 1)(𝑦 + 2) + 2𝑥(𝑥 + 2)
−𝑥(2𝑥 − 6) − (𝑥 + 3)(2𝑥 + 3)
A sidewalk was built around a rectangular swimming pool. Which expression below describes the area of the
sidewalk?
2x+5
3x+3
3x
x+4
Unit 8 – Polynomials 16
Practice: (𝑥 + 4)(𝑥 − 4)
(4𝑚 − 9)2
2𝑔(𝑔 − 5) + (𝑔 + 2)(3𝑔 − 1)
(5𝑧 + 2)(𝑧 − 9) − (𝑧 − 5)
Find the area of the shaded portion.
(𝑥 + 8)(𝑥 − 8)
(𝑦 − 6)2
2x
3x+5
12x+2
5x-2
Unit 8 – Polynomials 17
Unit 8 – Polynomials 18
Factoring: Greatest Common Factor
Distributive Property Factoring a GCF
𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 𝑎𝑏 + 𝑎𝑐 = 𝑎(𝑏 + 𝑐)
Think: “Un - _______________”
What is the ___________ number all the coefficients can divide by?
That’s the constant you can Factor out.
Do ALL the terms have a common variable?
o If so, what is the ___________exponent of that variable?
That’s how many you can Factor out.
Examples:
𝑥2 + 3𝑥
15𝑦3 + 20𝑦5 − 10
−8𝑠4𝑡 + 20𝑡3 − 28𝑡2
20𝑎𝑐2 + 15𝑎2𝑐 − 5𝑎𝑐
Unit 8 – Polynomials 19
Practice:
𝑥2 + 2𝑥
5𝑚3 + 45
10𝑥2 − 2𝑥
20𝑥2𝑦2 − 4𝑥𝑦
−5𝑚4𝑛 − 5𝑚3𝑛 + 5𝑚2
𝑏2 − 8𝑏
8𝑐2 + 7𝑐
44𝑎2 + 11𝑎
−12𝑡5 + 6𝑡
9𝑥2 + 6𝑥 + 18
Unit 8 – Polynomials 20
Factoring: Greatest Common Factor Cont…
Examples:
The area of a rectangle is 4x2 + 10x and the width is 2x. Find the measure of the length.
x(z – 5) + 4(z – 5) 3x(2x+1) – 1(2x+1)
Practice:
15 - 5a2
10g3 - 3g 15x4 - 5x2
8y(d – 2) + 3(d – 2)
x(z + 10) + 7(z + 10) x(x - 4) + 3(x - 4)
2r(r – 1) + 3(r – 1)
-20ab3c4d – 5ab2c2 3x3y - 6x2y2 - 3xy3
The area of a garden is 14x2y2z + 21xy2z2. The length is 7xy2z. What is the width?
Unit 8 – Polynomials 21
Unit 8 – Polynomials 22
Fatoring: Trinomials
Examples:
𝑥2 + 8𝑥 + 15
25 17 6x x
𝑥2 + 13𝑥 + 36
3x2 + 14x + 8
Unit 8 – Polynomials 23
Practice:
𝑥2 + 8𝑥 + 7
24 16 15x x
26 19 10x x
𝑥2 + 16𝑥 + 64
𝑥2 + 7𝑥 + 6
2x2 + 11x + 14
𝑛2 + 6𝑛 + 9
23 17 20x x
𝑛2 + 11𝑛 + 10
𝑥2 + 26𝑥 + 48
Unit 8 – Polynomials 24
Fatoring: Trinomials Cont…
Examples:
𝑥2 + 2𝑥 − 3
2x2 – 6x + 4
𝑎2 − 13𝑎 − 30
24 9 9x x
Unit 8 – Polynomials 25
Practice:
𝑥2 − 11𝑥 + 30
23 4x x
24 12 7x x
𝑥2 − 8𝑥 + 12
𝑚2 + 𝑚 − 90
25 7 6x x
7x2 - 5x – 2
𝑛2 + 4𝑛 − 12
Unit 8 – Polynomials 26
Fatoring: Special Answers
Examples:
x2 − 9
4𝑥2 − 25
4𝑥2 + 20𝑥 + 25
6x2 – 7x – 3
Unit 8 – Polynomials 27
Practice:
𝑥2 − 100
9𝑥2 − 4
9𝑥2 − 12𝑥 + 4
4x2 + 4x - 15
𝑥2 + 10𝑥 + 25
9𝑦2 − 121
6x2 – 13x – 15
8x2 – 2x – 1
Unit 8 – Polynomials 28
Fatoring: Trinomials with Greatest Common Factors
Sometimes you have to factor out a GCF and then continue factoring:
6𝑥2 + 6𝑥 − 36
6(𝑥2 + 𝑥 − 6)
(𝑥 + 3)(𝑥 − 2)
Factoring Review:
3𝑥2 + 15𝑥 + 18
23 8 5x x
4x2 – 16
4𝑥2 − 4𝑥 + 168
Unit 8 – Polynomials 29
Practice:
3𝑥2 + 9𝑥 − 120
24 4 1x x
5𝑥2 − 50𝑥 + 45
216 1x
23 8 4x x
22 18x
23 12 12x x
4x2 + 10x – 6
Unit 8 – Polynomials 30
Unit 8 – Polynomials 31