# 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.

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. $x^3 + 4x^2 + x + 4$