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. What is the definition of a term? How about a factor?
  2. When using the word "term" or "factor", what do you need to specify alongside the word?
  3. 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?
  4. In fraction property #5, which says \[\dfrac{ac}{bc} = \dfrac{a}{b}\] what does $c$ need to be in order to be cancelled out?
  5. 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.
  6. 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.
  7. 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.
  8. Your friend tries to simplify $(x + 3)(x + 2)$ by using the Distributive law. They treat $(x + 3)$ as a factor and distribute to $x$ and $2$: \[(x + 3)(x + 2) = x + 3 \cdot x + x + 3 \cdot 2\] What did they do incorrectly?
  9. In the global context, is the expression \[-x(x-2)(x+3) + 5y\] comprised of terms or factors?
  10. 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
  11. For each of the following sets, draw their real line representation.
    See Lecture Note 1.1 for examples.
    1. $(-\infty, 1) \cup (1, 2) \cup (2, \infty)$
    2. $(-\infty, -6]\cup (2, 10)$
    3. $(-10, -4]\cup (4, \infty)$
  12. A student tries to simplify \[\dfrac{x+3}{x + 2} \cdot 4 = \dfrac{x+3\cdot 4}{x+2} = \dfrac{x+12}{x+2}\] Why is this incorrect?
  13. Rewrite the following expressions so there are no negative exponents and simplify.
    See Lecture Note 1.2 for exponent law examples.
    1. $x^{-2}$
    2. $\dfrac{y}{(x-2)^{-1}}$
    3. $\dfrac{(x-2)(x-3)^{-2}}{5(3x-2)^{-4}}$
  14. Rewrite each root as an exponent and simplify.
    1. $\sqrt[2]{x^5}$
    2. $\sqrt[4]{(x+y)^3}$
    3. $\dfrac{\frac{2}{\sqrt[3]{x^4}}}{\frac{5}{\sqrt[3]{x+1}}}$
  15. Use exponent laws/fraction properties to simplify the following. Remember, simplify means to write in one fraction + no negative exponents.
    1. $x \cdot \sqrt{x}$
    2. $\dfrac{x}{\frac{1}{\sqrt{x}}}$
    3. $3(2x+1)\cdot (2x+1)^{-1/2}$
  16. 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?
  17. 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?
  18. 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?
  19. A student tries to simplify $(2x + \sqrt{x})^2$ by applying exponent laws. They write \[(2x + \sqrt{x})^2 = 2x^2 + x\] State the two errors they made and why they are incorrect.
  20. True or false: Like terms are expressions which share the same factors, except possibly for the coefficient (the number).
  21. 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 $4(4x^2+3)^2(3x-2)$
  22. Expand and simplify each expression by using the distributive law (or FOIL) and combining like terms.
    See Lecture Note 1.3 for expanding examples.
    1. $(2x^2 + 3x) + (3x^3 + 2x)$
    2. $(x+1)(x-2)$
    3. $(x^2 + 2x + 1)(x-2)$
    4. $(1 - x)^2$
    5. $(x^6 - x^5) -2(x^4 - x^3) - x(x^2 - x)$
    6. $3(x+h)^2 - 1 - (3x^2 - 1)$
  23. Factor the following expressions.
    See Lecture Note 1.3 for factoring examples.
    1. $-2x^3 - x^2$
    2. $(x+3)^2(x-2) + (x+3)(x-2)^2$
    3. $x^2 - 1$
    4. $x^2 + 5x + 6$
    5. $x^2 + 13x + 12$
    6. $2x^2 + 7x + 3$
    7. $2x^2(x-1) + 7x(x-1) + 3(x-1)$
    8. $4a^2 - 9b^2$
    9. $(x^2 + 1)^2 - 7(x^2 + 1) + 10$
    10. $x^3 + 4x^2 + x + 4$
  24. Suppose your professor tells you to simplify the expression \[\dfrac{\dfrac{2}{x-2} - \dfrac{3}{x - 3}}{1 + \dfrac{1}{x + 4}}\]
    1. What type of expression is this called?
    2. How should you think about approaching this type of expression?
  25. Perform the indicated operation and fully simplify (meaning write as one fraction only, get rid of negative exponents, remove all compound fractions).
    See Lecture Note 1.4 for fraction manipulation examples and compound fractions.
    1. $\dfrac{3}{10} + \dfrac{4}{15}$
    2. $\dfrac{\sqrt{x^5}}{x^{-1/5}}$
    3. $\dfrac{1}{x + 1} - \dfrac{1}{x - 1}$
    4. $\dfrac{x}{x + 3} - \dfrac{18}{x^2 - 9}$
    5. $\dfrac{1}{x-1} + \dfrac{1}{x} - \dfrac{1}{x-1}$
    6. $\dfrac{4(x+3)(x-1)}{2(x-1)}$
    7. $\dfrac{x^2 + 2x + 1}{x^2 - 1}$
    8. $\dfrac{x^2h + 2xh + h}{h}$
  26. Draw a coordinate plane and graph the points $(1,2), (2, -1), (-3, -4)$ and $(4, -3)$.
  27. Suppose I have an expression $f(x)$ and I find one input which gives two different evaluations. In particular, I find that $x = 2$ spits out $f(2) = 5$ and $f(2) = 3$. Is $f$ a function?
  28. Describe two examples of functions which you encounter in your daily life.
  29. Let $f(x) = x^2 - x + 1$. Evaluate and simplify the following:
    1. $f(1)$
    2. $f(-a)$
    3. $f(x + h)$
    4. $f(x + h) - f(x)$
    5. $f(-x^2)$
  30. Let $f(x) = \dfrac{1}{x + 1}$ Evaluate and simplify the following:
    1. $f(-1)$
    2. $f(x + h)$
    3. $f(x + h) - f(x)$
  31. Consider the expression $f(x)g(x)$.
    1. Suppose $g(x) = x^2 + x + 2$. Substitute $g(x)$ into the above expression correctly.
    2. Now suppose \[f(x) = -x+2 \qquad g(x) = x^2 - 3\] Substitute $f(x)$ and $g(x)$ into the above expression correctly and fully simplify.
    3. When substituting two or more terms into a (local/global) factor, what do you need to not forget?
  32. Let \[f(x) = \begin{cases} 3x + 2 & x \geq 2 \\ x^2 - x & x < 2\end{cases}\] Evaluate the following:
    1. $f(0)$
    2. $f(1)$
    3. $f(2)$
    4. $f(3)$
  33. Draw a rough sketch of the previous graph by hand. You may verify your graph in Desmos.
    Hint: To graph a piecewise function in Desmos, you need to specify a domain restriction.
    You can graph $3x + 2, x \geq 2$ by typing the symbols \[3x + 2 \{x \geq 2\}\] into Desmos, where the $\geq$ sign is created by first typing $>$ then $=$.
  34. Graph the function by hand: \[f(x) = \begin{cases} -x^2 & x > 0 \\ -x - 1 & x \leq 0 \end{cases}\]
  35. Graph the function by hand: \[g(x) = \begin{cases} 2x & x > 0 \\ x & x < 0 \\ 3 & x = 0\end{cases}\]
  36. Draw one curve in the plane that isn't the graph of a function.
  37. Suppose $f(x)$ and $g(x)$ are two different functions. In the expression $f(x)g(x)$, is $f(x)$ a term or a factor in the global context?