Homework 1

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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.

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Answer the following:

1. 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?
2. In fraction property #5, which says \[\dfrac{ac}{bc} = \dfrac{a}{b}\] what does $c$ need to be in order to be cancelled out?
3. 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.
4. 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.
5. 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.
6. 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?
7. How many factors is $2x$ comprised of?
8. In the global context, is the expression \[2x + 3y^2\] comprised of terms or factors?
9. In the global context, is the expression \[-x(x-2)(x+3)5\] 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.
    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 \[\dfrac{x^{-1} + y^{-1}}{4} = \dfrac{1}{4xy}\] Why are you not allowed to do this?
13. 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.
14. 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.
    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. $\dfrac{xy^{-3}z}{(2x)^{-1}y^2z^{-2}}$
    7. $\dfrac{(x+1)(x+2)}{(x+1)^{-2}(x+2)^2(x+3)}$
    8. $\dfrac{\sqrt[3]{x^2}}{x^{2/3}}$
    9. $\dfrac{(-3)^4\sqrt{x}}{3^2\sqrt[3]{x}}$
    10. $-2^4$
    11. $(-2)^4$
    12. $(2x-1)^{\frac{2}{3}}(2x-1)^{-\frac{1}{3}}$
    13. $(-3x)^2(-4x(x-1))^2$
    14. $\left(\dfrac{x^8y^{-2}}{(x-1)(x+2)^2}\right)^{-1/2}$
15. 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?
16. 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?
17. 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?
18. 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.
19. True or false: Like terms are expressions which share the same factors, except possibly for the coefficient (the number).
20. 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 $4(4x^2+3)^2(3x-2)$
21. Expand and simplify each expression by using the distributive law and combining like terms.
    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. $3(x+h)^2 - 1 - (3x^2 - 1)$
22. Factor the following expressions.
    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. $x^3 + 4x^2 + x + 4$