**Directions:**

- Show each step of your work and fully simplify each expression.
- Turn in your answers in class on a physical piece of paper.
**Staple multiple sheets together.**- Feel free to use Desmos for graphing.

Answer the following:

- In the expression \[-42(x^2 + 3)(x+3)^2 + 5(x+4)^4(x-3)^2\] in what context is the expression $(x^2 + 3)$
**not**considered a factor, even though it is visually next to a multiplication? - In fraction property #5, which says \[\dfrac{ac}{bc} = \dfrac{a}{b}\] what does $c$ need to be in order to be cancelled out?
- Can I cross out the $x^2$ in \[\dfrac{x^2 + 1}{x^2 + 2}\] to get $\dfrac{1}{2}$? Give the reason why or why not.
- Can I cross out the $x - 1$ in \[\dfrac{(x-1)(x+2)}{(x-1)(x+3)}\] to get $\dfrac{x+2}{x+3}$? Give the reason why or why not.
- Can I cross out the $x - 1$ and $x + 3$ in \[\dfrac{(x-1)(x+2) + 4(x+3)}{(x-1)(x+3)}\] to get $\dfrac{x+2 + 4}{1}$? Give the reason why or why not.
- Your friend tries to simplify $(x + 3)(x + 2)$ by using the Distributive law: \[(x + 3)(x + 2) = x + 3 \cdot x + x + 3 \cdot 2\] What did they do incorrectly?
- How many factors is $2x$ comprised of?
- In the global context, is the expression \[2x + 3y^2\] comprised of terms or factors?
- In the global context, is the expression \[-x(x-2)(x+3)5\] comprised of terms or factors?
- Write down
**one**fractional expression which satisfies the following:- Global context of numerator comprises of three terms
- Global context of denominator comprises of two terms
- Each term in the numerator contains two factors
- Each term in the denominator contains three factors

- For each of the following sets, draw their real line representation.
- $(-\infty, 1) \cup (1, 2) \cup (2, \infty)$
- $(-\infty, -6]\cup (2, 10)$
- $(-10, -4]\cup (4, \infty)$

- A student tries to simplify \[\dfrac{x^{-1} + y^{-1}}{4} = \dfrac{1}{4xy}\] Why are you not allowed to do this?
- A student tries to simplify \[\dfrac{x+3}{x + 2} \cdot 4 = \dfrac{x+3\cdot 4}{x+2} = \dfrac{x+12}{x+2}\] Which mathematical property was violated?

Hint: parentheses were forgotten. - Use exponent laws/fraction properties to simplify the following. Remember, simplify means to
**write in one fraction + no negative exponents.**

I advise calling out each exponent law as you use them to help discriminate the laws between each other.- $x^4 \cdot x^7$
- $\left(\dfrac{1}{2}\right)^{-1}$
- $(x^2y)^3$
- $\left[\dfrac{(x^3y^5)^4}{z^{-4}}\right]^0$
- $\left(\dfrac{x}{y}\right)^2 \cdot \left(\dfrac{y^2}{z}\right)^3$
- $\dfrac{xy^{-3}z}{(2x)^{-1}y^2z^{-2}}$
- $\dfrac{(x+1)(x+2)}{(x+1)^{-2}(x+2)^2(x+3)}$
- $\dfrac{\sqrt[3]{x^2}}{x^{2/3}}$
- $\dfrac{(-3)^4\sqrt{x}}{3^2\sqrt[3]{x}}$
- $-2^4$
- $(-2)^4$
- $(2x-1)^{\frac{2}{3}}(2x-1)^{-\frac{1}{3}}$
- $(-3x)^2(-4x(x-1))^2$
- $\left(\dfrac{x^8y^{-2}}{(x-1)(x+2)^2}\right)^{-1/2}$

- A student tries to simplify $x^2 + x^3$ by applying exponent laws. They write \[x^2 + x^3 = x^{2+3} = x^5\] Why is this incorrect?
- A student tries to simplify $x^2 \cdot x^3$ by applying exponent laws. They write \[x^2 \cdot x^3 = x^{2\cdot3} = x^6\] Why is this incorrect?
- A student tries to simplify $(a + b)^2$ by applying exponent laws. They write \[(a + b)^2 = a^2 + b^2\] Why is this incorrect?
- A student tries to simplify $(2x + \sqrt{x})^2$ by applying exponent laws. They write \[(2x + \sqrt{x})^2 = 2x + x\] State the two errors they made and why they are incorrect.
- True or false: Like terms are expressions which share the same factors, except possibly for the coefficient (the number).
- State whether each pair of expressions are like terms or not.
- $3x^2$ and $4y$
- $3x^2$ and $4x$
- $x^3y$ and $4x^3y$
- $5(x+1)(x+2)$ and $-(x+1)(x+2)$
- $-100(3x-2)(4x^2+3)^2$ and $4(4x^2+3)^2(3x-2)$

- Expand and simplify each expression by using the distributive law and combining like terms.
- $(2x^2 + 3x) + (3x^3 + 2x)$
- $(x+1)(x-2)$
- $(x^2 + 2x + 1)(x-2)$
- $(1 - x)^2$
- $(x^6 - x^5) -2(x^4 - x^3) - x(x^2 - x)$
- $3(x+h)^2 - 1 - (3x^2 - 1)$

- Factor the following expressions.
- $-2x^3 - x^2$
- $(x+3)^2(x-2) + (x+3)(x-2)^2$
- $x^2 - 1$
- $x^2 + 5x + 6$
- $x^2 + 13x + 12$
- $2x^2 + 7x + 3$
- $2x^2(x-1) + 7x(x-1) + 3(x-1)$
- $4a^2 - 9b^2$
- $(x^2 + 1)^2 - 7(x^2 + 1) + 10$
- $x^3 + 4x^2 + x + 4$