Multiplying and Dividing Polynomials

Unit 9: Polynomials

Topic #1: Dividing a Polynomial by a Monomial

Topic #2: Dividing a Polynomial by a Polynomial w./ Long Division

​Ps: I, Ms. Wiley, actually made this lesson, so please go all of the way through it. It's not short, but I'm hoping that it's what you need! I miss you guys already!

Assignment:

• complete this Quizizz lesson (I recommend taking notes, but I will not be checking to see if you have)

• complete the "🔵 HW/CW Unit 9 Dividing Polynomials #1" Quizizz in Google Classroom.

Learning Goals:

• be able to divide a Polynomial expression by a Monomial expression

• be able to divide polynomial expression by a polynomial expression using long division

Goals:

Please note: Tomorrow there will be a quiz on operations. I will include more information about the quiz at the end of this lesson.

Reviewing Other Operations

Here are the operations that we learned/practiced last week:

• addition

• subtraction

• multiplication

Let's practice these with a few questions first to warm-up. The exponent rules to the right will help hopefully help refresh your memory.

Multiple Choice

Find the sum.

(5x² -3x + 4) + ( -3x² +6x- 3)

• 3x² + 2x + 1

• 2x² + 3x +1

• 2x² + 5x - 7

• 3x² + 4x⁵ -7x

Multiple Choice

Find the difference.

(3- 2x + 2x²) - (3x²+4x-5)

• x² + 6x + 8

• 2x² + 5x - 7

• -x² -6x + 8

• -2x² + 11x - 4

Multiple Choice

Multiply the following:

(x-3)(x²+2x+7)

• x³-x²+x-21

• x³+5x+13x-21

• x³-3x²+7x-21

• x²+2x-3

Multiple Choice

Multiply the following:

7x²y³ (2x⁴y⁵+6xy³)

• 14x⁸y¹⁵+42x²y⁹

• 9x⁶y⁸+13x³y⁶

• 14x⁶y⁸+42x³y⁶

• 14x⁶y⁸+42x²y⁶

Multiple Choice

The following polynomial is written in standard form:
7x³ - 11x² + 8x - 5

• True

• False

Multiple Choice

What is the standard form of the polynomial?

 9x² + 5x + 27 + 2x³

• 27 + 5x + 9x² + 2x³

• 9x² + 5x + 2x³ + 27

• 9x² + 5x + 27 + 2x³

• 2x³ + 9x² + 5x + 27

Multiple Choice

Find the sum.
(3y² + y³ - 5) + (4y² -4y + 2y³ + 8)

• 3y³ + 7y² - 4y + 3

• 4y² + 3y³ - y + 3

• -y⁴ + 2y³ -y - 4

• 4y³ + 3y² -3y + 1

Multiple Choice

The largest exponent which is shown in a polynomial.

• exponent

• cubic

• degree

• linear

Reorder

Reorder the following polynomial into Standard Form

( 5 is the first number, 1 is the last number)

$$-x^2-15x+7x^4-6+3x^3$$  

• $$+7x^4$$  

• $$-6$$  

• $$-x^2$$  

• $$+3x^3$$  

• $$-15x$$  

1

2

3

4

5

• 1
• 2
• 3
• 4
• 5

Math Response

Simplify: Final Answer should be in standard form

$$\left(x^3+2x^2+5\right)+\left(3x^3+4x^2-2x-1\right)$$

Type answer here

Correct Answer: 4x^3+6x^2-2x+4

Deg^°

Rad

Math Response

Simplify: Final Answer should be in standard form:

$$\left(5x^3+6x^2-4x-1\right)-\left(2x^3-3x^2-4x+5\right)$$

Type answer here

Correct Answer: 3x^3+9x^2-6

Deg^°

Rad

Math Response

Simplify: Final answer should be in standard form

$$3x(4x^2+3x-1)$$

Type answer here

Correct Answer: 12x^3+9x^2-3x

Deg^°

Rad

Math Response

Simplify: Final answer should be in standard form

$$\left(2x+1\right)\left(2x-1\right)$$

Type answer here

Correct Answer: 4x^2-1

Deg^°

Rad

Multiple Choice

According to exponent rules, when we raise a power to a power we _______ the exponents.

• add

• subtract

• multiply

• divide

Multiple Choice

According to exponent rules, when we multiply like bases we _______ the exponents.

• add

• subtract

• multiply

• divide

Multiple Choice

According to exponent rules, when we divide like bases we _______ the exponents.

• add

• subtract

• multiply

• divide

The following slides contain videos that teach you how to simplify expressions that look like this ⬇

​

Topic #1: Dividing a Polynomial Expression by a Monomial Expression

​Here is a graphic that refreshes you on the rules of exponents that we covered in class on Friday. Different letters are used, but the rules are the same.

There will also be practice questions for you to build your understanding with.

​This video shows how to divide a monomial by a monomial with positive exponent results.

Multiple Choice

Evaluate the expression using the quotient rule.

• x¹³

• x⁵

• x³⁶

• x^-5

Multiple Choice

Evaluate the expression using the quotient rule.

• 1

• g¹⁰

• g²⁵

• g

Multiple Choice

25x⁵/5x³

• 5x

• 5x⁸

• -5x²

• 5x²

Multiple Choice

What is the answer to 10b/2b?

• 5b

• 4

• 5

• 4b

Multiple Choice

Simplify:  27a⁹ ÷ -3a³

• -9a³

• -9a⁶

• -24a³

• 24a⁶

Multiple Choice

(36x⁵y²) ÷ (-9x³y)

• -4x²y

• 4x²y

• -4x⁸y³

• 4x⁸y³

Multiple Choice

(-20x⁷y⁵) ÷ (-4x³y²)

• -5x⁴y³

• 5x⁴y³

• 5x¹⁰y⁷

• -5x¹⁰y⁷

​This video models dividing monomials that result in negative exponents.

Multiple Choice

• 2a²b⁶

• 1/2a²b⁶

• b⁶/2a²

• 2b⁶/a²

Multiple Choice

Multiple Choice

Simplify.

• x⁷

• ¹/_x3

• x³

• ¹/_x7

Multiple Choice

Write with positive exponents:

• a^-2

• a²

• 1/a¹

​This video covers dividing a polynomial by a monomial.

Multiple Choice

When I am a dividing by a monomial, I should _______ the coefficients and ________ the exponents for each term.

• add, subtract

• subtract , divide

• divide, divide

• divide, subtract

Multiple Choice

$$\frac{\left(8x^4+6x^3-10x^2\right)}{2x^2}$$  

• $$4x^2+3x-5$$  

• $$4x^2+3x^{1.5}-5x$$  

• $$6x^2+4x-12$$  

• $$6x^2+4x^{1.5}-12x$$  

Multiple Choice

$$\frac{\left(-18x^5-12x^3-6x^2\right)}{-3x}$$  

• $$-15x^4-9x^2-3x$$  

• $$-6x^4-4x^2-2x$$  

• $$6x^4+4x^2+2x$$  

• $$6x^5+4x^3+2x^2$$  

Multiple Choice

$$\frac{81x^9-54x^6+27x^3}{9x}$$  

• $$9x^9-6x^6+3x^3$$  

• $$9x^8-6x^5+3x^2$$  

• $$9x^{10}-6x^7+3x^4$$  

• $$81x^8-6x^5+3x^2$$  

Multiple Choice

$$\left(14d^3-21d^8\right)\div\ 7d^2$$  

• $$7d-21d^6$$  

• $$7d^{1.5}-14d^4$$  

• $$2d-3d^6$$  

• $$2d^{1.5}-3d^4$$  

Multiple Choice

$$\frac{\left(m^2n+m^6n^3\right)}{mn}$$  

• $$m^3n^2+m^7n^4$$  

• $$m^2n+m^5n^2$$  

• $$m+m^5n^2$$  

• $$m+m^6n^3$$  

​This video covers dividing polynomials by polynomials with the long division method. The video is a little long, but it's thorough and covers multiple different possible cases you could come across.

​Simplify using long division.

​Long division of polynomials is similar to long division of numbers.

​

​A few important things:

(1) Line up like terms on top of each other.

(2) Only divide by the leading term of the outside polynomial.

​(3) Your answer will be written as:

How can I divide polynomials?

MGSE9-12.A.APR.6 Rewrite simple rational expressions in different forms using long division.​

​

Multiple Choice

Find the quotient when (21x² - 52x + 32) is divided by (7x - 8).

• 3x + 4

• 3x + 3

• 3x - 4

• 3x + 2

Multiple Choice

Divide:

(4x² - 24x + 35) / (2x - 5)

• 2x - 7

• 2x - 12

• 2x + 12

• 2x + 7

Multiple Choice

Simplify using long division. 

$$\left(8b^3-17b^2-b+2\right)\div\left(b-2\right)$$  

• $$8b^2-2b-2-\frac{9}{b-2}$$  

• $$8b^2-2b-3-\frac{4}{b-2}$$  

• $$8b^2-2b-3-\frac{7}{b-2}$$  

• $$8b^2-b-3-\frac{4}{b-2}$$  

Multiple Choice

Simplify using long division.

$$\left(x^4-7x^2+8x-5\right)\div\left(x-2\right)$$  

• $$x^3+2x^2-3x+2-\frac{1}{x-2}$$  

• $$x^3+2x^2-4x+1+\frac{1}{x-2}$$  

• $$x^3+2x^2-4x$$  

• $$x^3+2x^2-4x+\frac{4}{x-2}$$  

About tomorrow's quiz...

What does it cover?

• Addition

• Subtraction

• Multiplication

• Division (monomials & polynomials w./ long division)

How many questions?

• 8 (3 on division)

What is it on?

• Edulastic w./ check answer turned on (2 checks per question. 2 attempts on quiz since I am not there to answer any questions.)

When is it?

• To be completed during class time, either before or after completing the Quizizz lesson for the day.