Homework 1

Directions:

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

Answer the following:

  1. State the property of real numbers shown below:
    1. $2 + 3 = 3 + 2$
    2. $2 \cdot 3 = 3 \cdot 2$
    3. $x(x+y) = x\cdot x + x\cdot y$
    4. $(a + b)(c + d) = (a + b)c + (a + b)d$
    5. $(3\cdot x)\cdot y = 3\cdot(x\cdot y)$
  2. 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 wrong?
  3. Let $x \in \mathbb{R}$. In English, what does $\lvert x \rvert$ mean?
  4. If \[A = \{1,2,3,4,5,6,7\} \qquad B = \{2,4,6,8\} \qquad C = \{7,8,9,10\}\] find the following:
    1. $A\cup B$
    2. $B\cup C$
    3. $A \cap C$
    4. $A \cup B \cup C$ (hint: solve $A \cup B$ first, then union that set with $C$.)
    5. $A \cap B \cap C$
  5. Simplify and graph the resulting set:
    1. $(-2, 0) \cup (-1, 1)$
    2. $(-2, 0] \cap (-1, 1)$
    3. $(-\infty, 6]\cap (2, 10)$
    4. $(-\infty, -4]\cup (4, \infty)$
  6. Simplify the following:
    1. $\dfrac{3}{10} + \dfrac{4}{15}$
    2. $\dfrac{3}{5} \div \dfrac{6}{5}$
    3. $2 \cdot \dfrac{1}{3} \cdot \dfrac{5}{7}$
    4. $\lvert 3 \rvert$
    5. $\lvert -3 \rvert$
    6. $\lvert - \lvert 3\rvert \rvert$
    7. $\lvert -\lvert -3\rvert \rvert$
    8. $\lvert 1 - \pi \rvert$
    9. $\lvert -2.76 \rvert$
  7. Use Exponent laws to simplify the following. It is not required but I advise calling out each exponent law as you use them to help discriminate the laws between each other.
    1. $x^4 \cdot x^7$
    2. $\left(\dfrac{1}{2}\right)^{-1}$
    3. $(x^2y)^3$
    4. $\left[\dfrac{(x^3y^5)^4}{z^{-4}}\right]^0$
    5. $\left(\dfrac{x}{y}\right)^2 \cdot \left(\dfrac{y^2}{z}\right)^3$
    6. (skip) $\dfrac{xy^{-3}z}{(2x)^{-1}y^2z^{-2}}$
    7. (skip) $\dfrac{(x+1)(x+2)}{(x+1)^{-2}(x+2)^2(x+3)}$
    8. (skip) $\dfrac{\sqrt[3]{x^2}}{x^{2/3}}$
    9. (skip) $\dfrac{(-3)^4\sqrt{x}}{3^2\sqrt[3]{x}}$
  8. For the expression \[(x + 3)(x-2)^2 + (x-4)(x-3) + 3\] Considering the context of the entire expression, what is
    1. $(x + 3)$
    2. $(x - 4)(x-3)$
    3. $(x - 3)$
    4. $(x+3)(x - 2)^2$
  9. In the expression \[-42(x^2 + 3)(x+3)^2 + 5(x+4)^4(x-3)^2\] why is $(x^2 + 3)$ usually not considered a factor (even though visually it is next to a multiplication)?
  10. In fraction property #5, which says \[\dfrac{ac}{bc} = \dfrac{a}{b}\] what does $c$ need to be in order to be cancelled out?
  11. 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.
  12. 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.
  13. 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.
  14. 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?
  15. 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?
  16. 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?

    17. The rest of these problems will be on next week's homework. Skip for now.

  17. You tried converting $\sqrt[5]{x^3}$ into $x^{\frac{5}{3}}$. Is this wrong?
  18. State whether each pair of expressions are like terms or not.
    1. $3x^2$ and $4y$
    2. $3x^2$ and $4x$
    3. $x^3y$ and $4x^3y$
    4. $5(x+1)(x+2)$ and $-(x+1)(x+2)$
    5. $-100(3x-2)(4x^2+3)^2$ and $4(4x^2+3)^2(3x-2)$
  19. Expand and simplify each expression.
    1. $(2x^2 + 3x) + (3x^3 + 2x)$
    2. $(x+1)(x-2)$
    3. $(x^2 + 2x + 1)(x-2)$
    4. $(1 - x)^2$
    5. $(x^6 - x^5) -2(x^4 - x^3) - x(x^2 - x)$
  20. Factor the following expressions.
    1. $-2x^3 - x^2$
    2. $(x+3)^2(x-2) + (x+3)(x-2)^2$
    3. $x^2 + 5x + 6$
    4. $x^2 + 13x + 12$
    5. $2x^2 + 7x + 3$
    6. $2x^2yz + 7xyz + 3yz$