Practice Test 76 - Algebra 2

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. Convert $2\,\text{cis}\,45^\circ$ to rectangular form $a+bi$. (Recall $r\,\text{cis}\,\theta=r(\cos\theta+i\sin\theta)$ and $a=r\cos\theta$, $b=r\sin\theta$.)
2. A function has values at consecutive integers: $F(0)=7$, $F(1)=12$, $F(2)=17$, $F(3)=22$. Based on growth over equal intervals ($\Delta x=1$), does this represent linear or exponential growth?
3. A polynomial function has zeros at $x = -2$ (multiplicity 1), $x = 1$ (multiplicity 3), and $x = 4$ (multiplicity 2). If the function has a negative leading coefficient, which statement about the graph's behavior is correct?
4. A printing company's weekly cost (in dollars) to produce $x$ posters is modeled by $C(x)=200+0.80x$. In this function, what does the parameter $200$ represent in context?
5. Use polynomial division to rewrite $\frac{x^3+2x^2-5x+3}{x-1}$ in the form $q(x)+\frac{r(x)}{x-1}$, where $\deg(r)<\deg(x-1)$.
6. Solve by completing the square: $x^2 + 4x - 5 = 0.$
7. A student says $x^2+4$ cannot be factored because it is a sum of squares. Over $\mathbb{C}$, use the difference of squares idea $x^2- (2i)^2$ to factor $x^2+4$ completely.​
8. The magnitude of an earthquake is measured using the Richter scale: $M = \log\left(\frac{I}{I_0}\right)$, where $I$ is the intensity and $I_0$ is a reference intensity. An earthquake measuring 6.5 on the Richter scale is followed by an aftershock measuring 4.5. What is the ratio of the main earthquake's intensity to the aftershock's intensity?
9. Rewrite $\frac{x^2-4}{x-2}$ in the form $q(x)+\frac{r}{x-2}$ (you may simplify first), with $\deg(r)<\deg(x-2)$.
10. Recognize the pattern and factor. The expression $x^3-27$ is a difference of cubes. Which is the correct factorization of $x^3-27$?​​​
11. A food truck plans to make tacos (x) and salads (y). Each taco takes 2 minutes to assemble and each salad takes 5 minutes. The truck has at most 180 minutes of prep time. For menu balance, they must make at least 10 salads. Also $x\ge 0$ and $y\ge 0$. Is the plan $(x,y)=(60,10)$ viable? (A plan is viable if it satisfies all constraints.)
12. The function $f(x)=x^2-4x+3$ (domain: all real numbers) is not one-to-one and fails the horizontal line test. Which restriction makes $f$ invertible by keeping only one side of the vertex?
13. Prove by composition that $f(x)=\dfrac{2x+3}{5}$ and $g(x)=\dfrac{5x-3}{2}$ are inverses by showing $f(g(x))=x$ and $g(f(x))=x$.​
14. Determine whether $f(x)=x^3-x$ is even, odd, or neither.​
15. Use the difference of cubes identity $a^3-b^3=(a-b)(a^2+ab+b^2)$ with complex numbers. Let $a=1+i$ and $b=1-i$. What is $a^3-b^3$?
16. A function $f(x) = |x^2 - 9|$ is defined for all real numbers. To make this function invertible by restricting to an interval where $f(x) \geq 7$, which domain restriction works?
17. Quadratics $ax^2+bx+c=0$ always have exactly 2 solutions in $\mathbb{C}$ counting multiplicity: either 2 distinct real (discriminant $>0$), 1 repeated real counted twice (discriminant $=0$), or 2 complex conjugates (discriminant $<0$). For $x^2+2x+5=0$, use the discriminant to determine the type and then find both solutions. Which option is correct?
18. Use the inverse relationship to simplify: $2^{\log_2(17)}$.​
19. For which value of $k$ will $\frac{3 + ki}{1 + 2i}$ result in a purely real number?
20. Use polynomial division (or simplify first if you notice a pattern) to express $\frac{x^2-4}{x-2}$ in the form $q(x)+\frac{r(x)}{x-2}$, where $\deg(r)<1$.

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