Total Review!

Total Review!

Now that we have more algebra under our belts, let's revisit some older topics and see how they fit in!

We'll also review topics just to keep them fresh.

Part I: Properties of Real Numbers

• commutativity

• associativity

• inverses

• identities

• distribution

• equality

Multiple Choice

The polynomial  $2x^2-11x+3x^5+4$   can be re-written in standard form as $3x^5+2x^2-11x+4$  . Which property of real numbers guarantees that these polynomials are the same?

• Associativity of addition

• Associativity of multiplication

• Commutativity of addition

• Commutativity of multiplication

Poll

When solving for x, $\frac{x+2}{4}+2=\frac{x}{2}$   can be re-written without fractions as $x+2+8=2x$  . Which properties of real numbers were used? Select ALL that apply.

• Multiplicative Inverses

• Distribution

• Equality of multiplication

• Commutativity of multiplication

Inverses VS Identities

Remember: Identities do not change the value of an expression

Example: x+0=x

Remember: Inverses are used to "undo" an operation and give an identity

Example: x-x=0

Multiple Choice

(Level 1) The equation $\left(2a+19\right)\cdot1=2a+19$  

represents the use of... 

• ...the multiplicative identity

• ...the multiplicative inverse

• ...associativity of multiplication

• ...commutative property of multiplication

Open Ended

(Level 2) What property is shown below in all steps when solving for x?

Step 1:

 $\alpha\left|x\right|^2-16b=5$ 

Step 2: 
 $a\left|x\right|^2=16b+5$  


Step 3:
 $\left|x\right|^2=\frac{16b+5}{a}$  

Open Ended

(Level 3) Explain the role of the inverse in solving for x in the literal equation $y=mx+b$  .

Hint: Use this format and fill in the blanks in your answer-

First we _____ , which is an example of the _____ _____.

Then we _____ , which is an example of the _____ _____. 

Part II: Classifying Real Numbers

• Differentiate Rational numbers and irrational numbers

• Identify differences between numbers that are in the sets of rational, integers, whole numbers, and natural numbers

• classify operations of numbers

Open Ended

(KEY concept) What is the difference between a rational number and an irrational number?

Poll

(Concept) True or false: 

All whole numbers are integers, but not all integers are whole numbers.

• True

• False

Multiple Select

(Level 1) What sets do the numbers -4, 5, and 10 ALL belong to? Select all that apply.

• Irrational Numbers

• Rational Numbers

• Integers

• Whole Numbers

• Natural Numbers

Poll

(Level 2) True or false: 

The result of $\sqrt{5}\cdot\sqrt{5}$  is rational, but the result of  $\sqrt{5}+\sqrt{5}$  is not.

• True

• False

Multiple Select

(Level 3) The result of $\frac{10\sqrt{27}}{\sqrt{3}}\cdot\frac{2}{5}$  belongs to which number sets? Seect ALL that apply.

• Irrational Numbers

• Rational Numbers

• Integers

• Whole Numbers

• Natural Numbers

Part III: Polynomial Operations

• Power and Product Rules

• Negative Exponent and Quotient Rules

• Addition/Subtraction/Multiplication of polynomials

Multiple Select

(Concept) When multiplying expressions involving coefficients and exponents, such as $\left(2x\right)\cdot\left(3x^3\right)$  , which two rules apply? 

(Hint: pick 2- one for coefficients and one for exponents)

• Add Coefficients

• Multiply Coefficients

• Raise Coefficients to a Power

• Add Exponents

• Multiply Exponents

Multiple Select

Which of the following are always true?

•  $\left(2x\right)^2=2x^2$  

•  $\left(3x\right)^2=6x^2$  

•  $\left(4x\right)^2=16x^2$  

•  $\left(5+x\right)^2=25+x^2$  

Open Ended

(KEY concept) From the previous question, $\left(5+x\right)^2\ne25+x^2$  .

How can we rewrite  $\left(5+x\right)^2$  to explain this and avoid making this mistake?

Multiple Choice

Which represents the simplified version of $\left(2x^2y^4z^3\right)^3$ ? 

•  $8x^6y^8z^9$  

•  $6x^5y^7z^6$  

•  $8x^5y^7x^6$  

•  $6x^6y^8z^9$  

Multiple Choice

Which of the following represents $\left(3x+y\right)^2$  ?

•  $3x^2+y^2$  

•  $9x^2+y^2$  

•  $6x^2+6xy+y^2$  

•  $9x^2+6xy+y^2$  

Multiple Choice

(Rule): For any values of x and n,  $x^{-n}$ is always equivalent to...

•  $-x^n$  

•  $\frac{1}{x^n}$  

•  $-\frac{1}{x^n}$  

•  $\frac{1}{x^{-n}}$  

Multiple Select

(Concept): When dividing expressions involving coefficients and exponents, such as $\frac{2x}{3x^3}$  , which two rules apply? 

(Hint: pick 2- one for coefficients and one for exponents)

• Subtract Coefficients

• Divide Coefficients

• Negate coefficients

• Subtract Exponents

• Divide Exponents

Multiple Choice

(Level 1): Which of the following represent $\frac{2}{5}x^4y^{-3}z$ with only positive exponents? 

•  $\frac{2x^4z}{5y^3}$  

•  $\frac{2x}{5y^{-3}z}$  

•  $\frac{2y^3}{5x^4z}$  

Multiple Choice

(Level 2): Which of the following represents $\frac{9a^2b^{-5}}{a^5b}$  as an expression with single variables and only positive exponents

•  $9a^7b^{-4}$  
  

•  $9a^3b^6$  
  

•  $-\frac{9}{a^3b^5}$  

•  $\frac{9}{a^3b^6}$  

Poll

(Level 3): Write the expression below using single variables to positive exponents.

 $\frac{25x^{-4}y^6z}{5x^{-7}y^{-2}z^2}$  

•  $\frac{5y^4z^3}{x^{11}}$  

•  $\frac{5x^3y^8}{z}$  

•  $\frac{5y^8}{x^{11}z^2}$  

Multiple Choice

Find the Difference when (-2x³ + x² -20) is subtracted from

(4x³ -3x² + 6x - 4).

• 4x² - 6x³ - 2x + 6

• 3x + 5x² + 1

• 6X³ -4X² + 6X -2

• 3x + 2

Multiple Choice

Determine which expression represents  $A+B$  , where $A=(6x+4x^4-3x^2)$  and  $B=(7x^4+5x^2+8x)$  .

• 11x⁴ + 2x² + 14x

• 16x⁴ + x² + 14x

• 11x⁴ + x² + 14x

• 11x⁴ + 2x² + 10x

Multiple Choice

Which of the following represent the result when simplifying the expression below?

 $2a^2(3a-5)-6a(8a^2-2a+4)$  

• -42a³ + 2a² - 24a

• 6a³ - 10a²

• -48a³ + 12a² - 24a

• -42a³ - 22a² + 24a

Multiple Choice

Which of the following represents (4x+1)(5x−2) in standard form?

• 20x² + 3x − 2

• 20x² − 11x + 5

• 20x² − 18x + 2

• 20x² − 3x − 2

Multiple Choice

Find the product: (5n-8)(5n+8). Think: How is this different from (5n-8)²?

• 25n²-64

• 10n²

• 5n²+16

• 16

Multiple Choice

Simplify:
(3x - 5) ( 4x² - 2x - 3)

• 12x³ - 26x² - 25x + 15
• 12x³ - 20x² - 5x + 15
• 12x³ - 20x² - 6x + 15
• 12x³ - 26x² + x + 15

Part IV: Solving Equations

• Multi-step equations

• Fractional equations

• Literal Equations

Open Ended

What is always the goal when solving an equation for a given variable?

Multiple Choice

(Level 1) Which of the following represent a value of y that satisfies  $7(1-y)=-3(y-2)$   

• y = 1/4

• y = 4

• y = 5

Multiple Choice

(Level 2) Solve the equation below for x: 
 $-8x-8+3(x-2)=-3x+2$  

• x = -8

• x = 8

• x = 2

• x = -2

Multiple Choice

(Level 2) Which value of a will make the equation  $\frac{5a\ -\ 2}{3}\ =\ 3\ -\ a$   true? 

•  $\frac{8}{11}$  

•  $\frac{11}{8}$  

• 8

• 11

Multiple Choice

(Level 3) Solve the equation below for x.


 $\frac{2x\ +\ 1}{3}\ -\ \frac{1\ -\ x}{6}\ =\ -2$  

•  $\frac{-5}{13}$  

•  $-13$  

•  $-5$  

•  $\frac{-13}{5}$