Unit 9 Notes: Polynomials and Factoring

Algebra 1 Unit 9 Notes: Polynomials and Factoring

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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

http://www.mathguy.us/

http://www.khanacademy.com/

http://www.washoeschools.net/DRHSmath


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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


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Adding polynomials

Example 1: (4𝑥3 + 𝑥2 − 5) + (7𝑥 + 𝑥3 − 3𝑥2)

Example 2: Find the sum: (𝑥2 + 𝑥 + 8) + (𝑥2 − 𝑥 − 1)

Subtracting polynomials

Example 3: Find the difference: (4𝑧2 − 3) − (−2𝑧2 + 5𝑧 − 1)

Example 4: Find the difference of (3𝑥2 + 6𝑥 − 4) − (𝑥2 − 𝑥 − 7)

Example 5: You try! Simplify the expression: (3x2 + 5) – (x2 + 2) + (– 3m + 1)

Remember to

multiply each

term in the

polynomial by – 1

when you write

the subtraction

as addition.


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Multiplying polynomials by monomials

Example 6: Find the product 3𝑥3(2𝑥3 − 𝑥2 − 7𝑥 − 3)

Example 7: Multiply: −𝑥2(𝑥 − 6) Example 8: Simplify: 3𝑦3(𝑦 − 4)

Example 9: An online store purchases boxes to ship their products. The large box has a volume of 4𝑥3 +𝑥2 + 5 units. The medium box has a volume of 2𝑥3 + 3𝑥 − 4 units. The store purchases one large box and

two medium boxes. What polynomial expression represents the total volume of the purchased boxes?


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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 (5𝑥2 − 3𝑥 + 7) − (3𝑥 − 4𝑥2) + (2𝑥2 + 9 − 5𝑥3)

= 5𝑥2 − 3𝑥 + 7 − 3𝑥 − 4𝑥2 + 2𝑥2 + 9 − 5𝑥3

=−5𝑥3 + 3𝑥2 − 6𝑥 + 16

Christie (5𝑥2 − 3𝑥 + 7) − (3𝑥 − 4𝑥2) + (2𝑥2 + 9 − 5𝑥3)

= 5𝑥2 − 3𝑥 + 7 − 3𝑥 + 4𝑥2 + 2𝑥2 + 9 − 5𝑥3

=11𝑥4 − 5𝑥3 − 6𝑥2 + 16

Mark (5𝑥2 − 3𝑥 + 7) − (3𝑥 − 4𝑥2) + (2𝑥2 + 9 − 5𝑥3)

= 5𝑥2 − 3𝑥 + 7 − 3𝑥 + 4𝑥2 + 2𝑥2 + 9 − 5𝑥3

=−5𝑥3 + 11𝑥2 − 6𝑥 + 16

Example 11: Which of the following expressions is equivalent to 1

2𝑦2(6𝑥 + 2𝑦 + 12𝑥 − 2𝑦)?

A. 9𝑥𝑦2

B. 18𝑥𝑦

C. 3𝑥𝑦2 + 6𝑥

D. 9𝑥𝑦2 − 2𝑦3

E. 3𝑥𝑦2 + 12𝑥 − 𝑦3 − 2𝑦

Example 12: Find h(x) = f(x) + g(x) if 𝑓(𝑥) = (7𝑥2 − 3𝑥 + 2) and 𝑔(𝑥) = (5𝑥 – 2)

Example 13: Find h(x) = f(x)− g(x) if 𝑓(𝑥) = (−2𝑥3 − 4𝑥 + 2) and 𝑔(𝑥) = (5𝑥3 + 5𝑥2 – 2𝑥)


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9.2: Multiplying Polynomials

Warm-Up: Simplify each expression.

1) −3𝑥3(2𝑥2 − 9𝑥) 2) (2𝑠3 − 𝑠2 + 1) − (3𝑠2 − 𝑠 + 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) (𝒙 + 𝟑)(𝒙 + 𝟒)

b) (𝒙 + 𝟑)(𝒙 − 𝟐)


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c) Multiply: (𝟑𝒙 + 𝟕)(𝒙 − 𝟖)

d) Multiply: (𝒙𝟐 − 𝟒)(𝒙 − 𝒙𝟐)

Example 2: Find h(x) = f(x) ∙ g(x) if 𝑓(𝑥) = (2𝑥 + 7) and 𝑔(𝑥) = (𝑥 – 9).

Example 3: Multiply each expression.

a) (𝑥 + 4)2 b) (𝑥 − 7)2

c) (3𝑥 + 4)2 d) (5𝑥 − 1)2


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Example 4: Simplify the expression: (𝑚 + 7)(𝑚 – 3) + (𝑚 – 4)(𝑚 + 5)

Example 5: Find each product.

a) (𝑥 − 5)(𝑥 + 5) b) (𝑦 − 3)(𝑦 + 3) c) (2𝑎 − 7)(2𝑎 + 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 𝑥2 − 4.


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Example 7: Multiply the polynomials.

a) (2𝑎– 5)(𝑎2– 6𝑎– 3) b) (5p - 2)(3p2 – 2𝑝 + 1)

Example 8: You try! Find each product.

a) (3𝑥 − 2)(4𝑥2 − 5𝑥 + 1) b) (2𝑎2 + 3𝑎 − 2)(𝑎 − 4)

Example 9: Find h(x) = f(x) ∙ g(x) if 𝑓(𝑥) = (𝑥2 − 4𝑥 + 7) and 𝑔(𝑥) = (3𝑥 – 2)

Example 10: Simplify: 2(−4𝑎 + 9)2 + 5


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Example 11: If 𝑓(𝑥) = 𝑔(𝑥) and 𝑓(𝑥) = −3(𝑥 + 1)2 + 4, then what polynomial represents 𝑔(𝑥)?

Example 12: Which option below has a product of 𝑥2 − 4𝑥 + 4? Choose all that apply.

A) (𝑥 + 2)(𝑥 − 2)

B) (𝑥 − 2)(𝑥 − 2)

C) (𝑥 − 2)2

D) 2(𝑥2 − 𝑥)

Example 13: What would you have to multiple (𝑥 − 3) by to have a product of 𝑥2 − 9?

A) (𝑥 − 3)

B) (𝑥 − 9)

C) (𝑥2 − 3)

D) (𝑥 + 3)

Example 14: Which of the following expressions are equivalent to −𝑥2 + 3𝑥 + 28? Select all that apply.

A) (𝑥 + 4)(𝑥 − 7)

B) −(𝑥 + 4)(𝑥 − 7)

C) (𝑥 + 4)(7 − 𝑥)

D) (−𝑥 − 4)(𝑥 − 7)

E) (−𝑥 − 4)(7 − 𝑥)


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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?

−8𝑥 −20𝑥3 −10𝑥2

The GCF (Greatest Common Factor) is the largest common factor of two or more terms.

Factoring an expression by using the GCF:


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Examples 1 – 6: Factor each expression by taking out the GCF.

1) 5x + 20 2) 8x – 4x2

3) 16𝑥2𝑦 + 40𝑥y + 8𝑥𝑦2 4) −6𝑚2 − 30𝑚3

5) −12𝑎𝑏 + 32𝑏 6) −8𝑎𝑥3 + 𝑎𝑥2 − 3𝑎𝑥

Examples 7 – 10: Factor each expression by taking out the GCF.

7) −4𝑛𝑚 − 2𝑛2 8) −5𝑤𝑥3 + 10𝑤𝑥2

9) 6𝑦 − 15𝑦3 10) −9𝑑𝑚3 + 𝑑𝑚2 – 2𝑑4𝑚

11) One factor of −7𝑥3𝑦 − 21𝑥2𝑦2 is (−7𝑥2𝑦). What is the other factor?


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12) Factor: 15𝑥3 − 7𝑦4 − 2𝑧

For #13 – 15: Find each product. Try to do this without showing any work!!

13) (𝑥 + 2)(𝑥 + 3) 14) (𝑦 + 4)(𝑦 + 7) 15) (ℎ − 3)(ℎ + 5)

Challenge! What are the factors of each trinomial? (Try to work backwards to figure this out!)

16) 𝑥2 + 6𝑥 + 8 17) 𝑥2 + 7𝑥 + 10

Example 18: Write a polynomial expression to represent the area of the rectangle shown below, if 𝐴 = 𝑏ℎ.

Example 19: A rectangle has an area that can be represented by (3𝑥3 + 9𝑥2 + 6𝑥) 𝑓𝑡2. If the height of the

rectangle is 3𝑥 𝑓𝑡, then what expression can represent the base?


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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) (𝑥 − 3)(𝑥 + 3) 2) (x + 3)(x – 4)


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Work in groups to multiply (expand) the following expressions:

(𝑥 + 5)(𝑥 − 3) (𝑥 + 2)(𝑥 + 8) (𝑥 + 4)(𝑥 − 4)

Factoring a trinomial is the inverse (opposite) of multiplying binomials.

Example 1: Factor x2 + 10x + 16 Check by multiplying your answer:

Examples 2 – 4: Factor each expression.

2) x2 + 10x + 9 3) a2 – 6a + 9 4) 𝑥2 + 7𝑥 − 30

You try! Examples 5 – 7: Factor each expression.

5) 𝑥2 + 10𝑥 + 9 6) 𝑦2 − 𝑦 − 6 7) 𝑥2 + 2𝑥 + 1


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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 – 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 – 12: Factor each expression completely.

9) −𝑥2 + 4𝑥 + 12 10) 𝑤3 − 10𝑤2 + 25𝑤

11) −2𝑥2 + 14𝑥 − 24 12) 2𝑎2𝑏 − 10𝑎𝑏 + 8𝑏

Do you remember how to multiply conjugates? Multiply each expression below. (Try to do this

without work!)

(𝑥 − 5)(𝑥 + 5) (𝑥 + 11)(𝑥 − 11)

Factoring Difference of Two Perfect Squares:


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For #13 – 16: Factor each expression.

13) 𝑥2 − 25 14) 𝑎2 − 49𝑏2 15) 36 − 𝑦2 16) 𝑥6 − 100

You try! For #17 – 20: Factor each expression.

17) 𝑔2 − 4 18) 1 − 𝑏2 19) 𝑘2 − 81𝑗2 20) 𝑛10 − 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 – 29: Factor each expression completely.

21) 5𝑥2 − 20 22) −2𝑥3 + 32𝑥 23) 4𝑎5 − 4𝑎3

24) 𝑔2 + 16 25) −6𝑎4 + 36 26) −𝑥5 + 9𝑥3

27) 3𝑥2 − 24𝑥 + 12 28) −𝑥3 + 6𝑥2 + 16𝑥 29) 𝑎5 + 3𝑎4


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9.5: More Factoring Trinomials and Binomials

Warm-Up: Simplify each expression. Try to do these without showing work!

𝟏) (𝟑𝒙 − 𝟏)(𝒙 + 𝟒) 2) (𝟓𝒙 + 𝟐)(𝟑𝒙 − 𝟕)

Factoring Trinomials with a leading coefficient different than one:

Example 1: Factor 2𝑥2 − 11𝑥 + 5 Check your solution by using multiplication:


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2) 3n2 + 2n - 8 3) 2y2 – 13y – 7


4) 9𝑦2 + 6𝑦 + 1 5) 6𝑥2 + 5𝑥𝑦 − 6𝑦2 6) 15𝑥2 − 𝑥 − 6

You Try! Examples 7 – 9: Factor each expression.

7) 3𝑥2 − 5𝑥 + 2 8) 8𝑥2 + 14𝑥 − 15 9) 2𝑚2 + 𝑚𝑛 − 21𝑛2

Factoring Binomials with a leading coefficient different than one:


10) 25𝑥2 − 4 11) 49𝑏4 − 9𝑑2 12) 36𝑎2 − 𝑏6


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You try! For examples 13 – 15, factor each expression.

13) 121ℎ2 − 4𝑔8 14) 25 − 16𝑘2 15) 169𝑥2 − 49𝑦12

If you are able to factor out a GCF from an expression, always do that first!

Examples 16 – 18: Factor each expression completely. (Hint: look for a GCF first!)

16) 6𝑥2 − 2𝑥 − 4 17) 36𝑎5 − 9𝑎3 18) −4𝑥3 + 4𝑥2𝑦 + 3𝑥𝑦2

You Try! For examples 19 – 21, factor each expression completely.

19) 28𝑥2 + 38𝑥𝑤 − 6𝑤2 20) −6𝑥2 + 12𝑥 + 90 21) 25𝑥4 − 100𝑥2


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9.6 Notes: Factoring Completely

Warm-Up

1) Factor: 𝑥2 + 5𝑥 + 6 2) Factor: 𝑥2 − 64

3) Simplify: (𝑥 + 7)2 4) Simplify: (4𝑥2 − 3𝑥 + 7) − (7𝑥2 − 6𝑥 + 2)


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Examples 1 – 4: Factor each polynomial completely.

1) 5𝑎4– 405 2) 2𝑥2 − 8𝑥 − 10

3) 𝑥4 − 16 4) −𝑥3 − 𝑥2 + 12𝑥

Examples 5 – 10: Factor completely.

5) 3𝑟3 − 21𝑟2 + 30𝑟 6) 81𝑑5 − 𝑑 7) 2𝑥2 + 5𝑥𝑦 + 2𝑦2

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.


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8) 2y4 – 32 9) 49y2 – 25w6 10) −𝑥3 − 2𝑥2 + 15𝑥

11) Given (𝑥 + 4) is a factor of 2𝑥2 + 11𝑥 + 2𝑚, determine the value of m.

12) Which of the following expressions are equivalent to −𝑥2 + 4𝑥 + 21? Select all that apply.

A) (𝑥 + 3)(𝑥 − 7)

B) −(𝑥 + 3)(𝑥 − 7)

C) (𝑥 + 3)(7 − 𝑥)

D) (−𝑥 − 3)(𝑥 − 7)

E) (−𝑥 − 3)(7 − 𝑥)


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Work with your group to match each polynomial to its factors.

Polynomials

1. 8𝑥2 − 63𝑥 − 81

2. 49𝑥2 − 25

3. 8𝑥2 − 2𝑥 − 45

4. 2𝑥2 + 11𝑥 + 12

5. 𝑥2 − 7𝑥 − 8

6. 𝑥2 + 6𝑥 − 27

7. 𝑥2 − 4

8. 7𝑥2 + 32𝑥 − 60

Factors

A. (𝑥 + 9)

B. (7𝑥 − 5)

C. (𝑥 − 2)

D. (4𝑥 + 9)

E. (8𝑥 + 9)

F. (7𝑥 + 5)

G. (𝑥 − 9)

H. (2𝑥 + 3)

I. (7𝑥 − 10)

J. (𝑥 + 6)

K. (𝑥 − 3)

L. (2𝑥 − 5)

M. (𝑥 + 2)

N. (𝑥 − 8)

O. (𝑥 + 1)

P. (𝑥 + 4)

Are you done? Get your answers checked off, and then complete the Factoring Card Match with a partner.

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Unit 9 Notes: Polynomials and Factoring

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