Algebra II Polynomial

Intro to Polynomials

Classify and Simplify Polynomials

What is a Polynomial?

A polynomial is an expression involving a sum of powers in one or more variables multiplied by coefficients, where the powers must be whole.

What it IS and what it is NOT:

Adding Polynomials

1. Place like terms together
2. Add the like terms

Example: Adding Polynomials

Multiple Choice

Add the polynomials (x-2) and (x²+3)

• x²+x+1
• x³+5
• x³+1
• x³-1

Multiple Choice

Add these polynomials:

(5x + 2) + (4x + 5)

• 9x -7

• -9x-11

• 9x+7

• x+7

Multiple Choice

(2x + 5y - z) + (-6x - 4y + 7z)

• -4x +6y + 6z

• 4x + y + 6z

• -4x - y + 6z

• -4x + y + 6z

Subtracting Polynomials

To subtract Polynomials, first reverse the sign of each term we are subtracting (in other words turn "+" into "-", and "-" into "+"), then add as usual.

OR

Use distributive property with negative one on the second polynomial, then combine like terms.

Subtracting Polynomials Example

Multiple Choice

Subtract these polynomials:

(4x²+ x) - (x²+ 2x)

• 3x² - x

• 4x² + 2x

• 3x² + 3x

• 3x² + 2x

Multiple Choice

Subtract these polynomials:

(4n⁴ - 8n + 4) - (8n² + 4n⁴ + 1)

• -8n² - 8n + 3

• -7n² - 8n + 3

• -6n² - 8n + 3

• -7n² - 4n + 3

Multiple Choice

Subtract these polynomials:

(x⁵ + x³) - (6x - x³ + 6x⁵)

• -5x⁵ + 2x³ - 6x

• 7x⁵ - 6x

• -5x³ + 2x² - 6x

• -5x⁵ - 6x

Multiple Choice

Which expression represents the perimeter of the triangle?

• 8xy + 3

• 5x + y + 3

• 5x + 4y + 3

• 6x + 3y + 3

Multiple Choice

• 7x + 5

• 24x + 4

• 9x + 5

• 9x² + 5

Multiple Choice

If a rectangular field has a length of $$x+7$$  and a width of  $$4x-2$$  ,  write an expression for the length of fencing to go around the outer edges.

• $$5x+5$$  

• $$10x+10$$  

• Why do we need fences?

• $$4x^2-14$$  

Multiple Choice

What is the perimeter of the rectangle?

• x²+8x-5

• 2x²+16x-10

• 2x²+10x-8

• 6x-2

Factoring with GCF (Greatest Common Factor)

Factor out the greatest number and the smallest variable that each have in common.

Finding the GCF

• Find the largest number that divides evenly into all of the terms coefficients

• If all of the terms share the same variable(s) use those with the smallest exponent

Multiple Choice

GCF of 35f + 40f

• 5f
• 10
• f
• 2f

Multiple Choice

Factor:
2n²+16n+12

• 2n(n²+24n+6)
• 2(n²+8n+6)
• 2n(n²+8n+6)
• 3(n²+8n+6)

Multiple Choice

Factor by using GCF:
10x² + 15x-5

• 25(10x³+45x-15)
• 5(2x³ + 3x²-x)
• x(10x³+15x-5)
• 5(2x²+3x-1)

Multiple Choice

What is the greatest common factor of 4xy and 20y?

• 4y

• 20xy

• 2y

• xy

Multiple Choice

Factor the following polynomial by the GCF.

$$25x^3-30x^2+5x^{ }$$  

• $$5x\left(5x^2-6x+1\right)$$  

• $$x\left(25x^2-30x+5\right)$$  

• $$5x\left(\left(5x^2-6x\right)\right)$$  

• $$5x^2\left(5x-6\right)$$  

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

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

• $$4x^2+3x-5$$  

• $$4x^2+3x^{1.5}-5x$$  

• $$6x^2+4x-12$$  

• $$6x^2+4x^{1.5}-12x$$  

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

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

• $$7d-21d^6$$  

• $$7d^{1.5}-14d^4$$  

• $$2d-3d^6$$  

• $$2d^{1.5}-3d^4$$  

Multiplying Polynomials

Multiplying polynomials is like distributive property; whether you distribute once, twice, or three times.

After you distribute, combine like terms.

Multiple Choice

Simplify:

$$5\left(3x+2\right)$$  

• $$3x+10$$

• $$15x+10$$  

• $$15x-10$$  

• $$3x-10$$  

Multiple Choice

Simplify:
3a(5a² + 8a + 2)

• 15a³ + 24a² + 6a

• 10a³ + 24a + 6

• 8a³ + 11a² + 6a

• 15a³ + 24a + 6

Multiple Choice

Multiply

Multiple Choice

Multiply

Multiple Choice

4m(3m⁷ + 2m⁵)

• 7m⁷ + 6m⁶

• 12m⁸ + 8m⁶

Multiplying Polynomials

January 20 - 10B

Multiple Select

$$\left(x\ +\ 4\right)\left(x\ +\ 8\right)$$  Multiply:

• $$x^2\ +\ 12x\ +\ 32$$  

• $$32x^2$$  

• $$x^2\ +\ 32$$  

Multiple Select

$$\left(2x\ +\ 3\right)\left(3x\ +\ 2\right)$$  

• $$6x^2\ +\ 13x\ +\ 6$$  

• $$6x^2\ +\ 6$$  

• $$5x^2\ +10x\ +5$$  

Multiple Select

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

• $$2x^2\ +3x\ -20$$  

• $$2x^2\ -\ x\ -20$$  

• $$x^2\ -\ 20$$  

• $$x^2\ +\ 13x\ -20$$  

Multiple Choice

Simplify:
(x-7)²

• x²-49

• x²+49

• x²-14x+49

• x²-14x-49

Multiple Choice

Simplify:
(x - 7)(x + 7)

• x² + 7x - 49

• x² - 7x + 49

• x² - 49

• x² + 49

Multiple Choice

Find the product:
(x + 5)(x - 4)

• x² + 9x + 20

• x² + 9x - 20

• x² + 1x - 20

• x² - 1x - 20

Multiple Choice

(x + 5)(x + 4)

• x² + x + 20

• x² + 20x + 9

• x² + 9x + 20

• x² - 9x + 54

Multiple Choice

Find the product:

$$\left(5s+2\right)^2$$  

• $$25s^2+4$$  

• $$25s^2+10s+4$$

• $$25s^2+20s+4$$  

• $$5s^2+20s+4$$  

Multiple Choice

Multiply and Simplify.

$$\left(4x+1\right)\left(4x-1\right)$$  

• $$16x^2-1$$  

• $$8x^2-1$$  

• $$16x^2+1$$  

• $$8x^2+1$$  

Multiple Choice

Multiply and Simplify.

$$\left(3x+2\right)\left(2x-7\right)$$  

• $$6x^2-14$$  

• $$5x^2-5$$  

• $$6x^2-17x-14$$  

• $$6x^2+25x-14$$  

Multiple Choice

Multiply and Simplify.

$$\left(x+3\right)\left(x-5\right)$$  

• $$x^2-2x-15$$  

• $$x^2+3x-15$$  

• $$x^2-15$$  

• $$x^2+15$$  

Multiple Select

$$\left(x\ +\ 4\right)\left(x\ +\ 8\right)$$  Multiply:

• $$x^2\ +\ 12x\ +\ 32$$  

• $$32x^2$$  

• $$x^2\ +\ 32$$  

Multiple Choice

Using the rectangle, which expression represents the area (area = length times width)? 

• $$x^2-4x-12$$  

• $$2x^2-12$$  

• $$x^2+4x\ -12$$  

• $$2x-4$$