Practice Test 75 - Algebra 2

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

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1. A savings plan deposits $100 at the end of each month into an account earning $1\%$ per month (so the balance grows by a factor of $1.01$ each month). After $12$ deposits, the value is $100(1.01)^{11}+100(1.01)^{10}+\cdots+100(1.01)^0.$ What is the value after the 12th deposit (rounded to the nearest cent)?
2. Convert $5\left(\cos 270^\circ+i\sin 270^\circ\right)$ to rectangular form $a+bi$ using $a=r\cos\theta$ and $b=r\sin\theta$.
3. To sketch the parabola $y=x^2-25$, first factor to reveal the zeros. What are the zeros and corresponding x-intercepts?
4. Subtract and simplify to lowest terms: $\frac{x+1}{x^2-4}-\frac{3}{x-2}$ (Assume $x\neq 2,-2$.)
5. A preview of the Fundamental Theorem of Algebra says: every polynomial of degree $n$ has exactly $n$ complex zeros counting multiplicity. For quadratics ($n=2$), this means any $ax^2+bx+c=0$ has exactly 2 complex solutions counting multiplicity. Which statement best matches this quadratic-specific conclusion?
6. A sequence is defined by $s_2 = 7$, $s_3 = 11$, and $s_n = s_{n-1} + s_{n-2}$ for $n \geq 4$. As a function $f(n) = s_n$, what is the most restrictive description of the domain?
7. Interpret by viewing numerator and denominator as single entities: $\frac{x^2 + 3x - 2}{x - 1}$. Which statement correctly describes what depends on $x$ and what chunking helps you see (without doing long division)?
8. In the rational expression $\dfrac{3x^2 - 2x + 5}{x-1}$, identify the terms in the numerator.
9. A company's profit $P$ (in thousands of dollars) from selling $x$ units of a product is given by $P(x) = -2x^2 + 16x - 24$. The company wants to determine the range of units that must be sold to achieve a profit of at least $8,000. Which inequality represents this situation?
10. Use the Binomial Theorem (coefficients from Pascal's Triangle row 4: $1,4,6,4,1$) to expand $(x+2)^4.$
11. The polynomial $f(x) = x^8 - 256$ can be factored completely over the complex numbers into linear factors of the form $(x - r)$ where $r$ represents the 8th roots of 256. How many of these roots are purely imaginary (have zero real part)?
12. A runner's distance $d$ (in meters) after $t$ seconds is shown. $t$: 0, 2, 4, 6, 8 $d$: 0, 6, 12, 18, 24 Calculate $\Delta d/\Delta t$ for each interval and determine whether the rate of change is constant.
13. A geometric series is the sum of terms from a geometric sequence. Find the sum of the first $6$ terms of the geometric series $3+6+12+24+48+96.$
14. A sequence follows the pattern: 5, 20, 80, 320, ... . If this pattern continues, what are both the recursive and explicit formulas for the nth term, where $a_1 = 5$?
15. Consider the functions (for $x\ge 0$): linear $f(x)=50x$, quadratic $g(x)=2x^2$, polynomial $h(x)=x^5$, and exponential $p(x)=1.2^x$. Which statement about long-term behavior is true?
16. Is the rate of change constant for the function $f(x)=5x-2$? Explain in terms of linear functions and slope.
17. Function $f$ is given by $f(x)=x^2-6x+5$. Function $g$ is shown in the table: $\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4\\ \hline g(x) & 2 & 0 & -2 & 0 & 2 \end{array}$ Which statement correctly compares the minimum values of $f$ and $g$?
18. A ball's height is modeled by $y=-x^2+6x$ and a laser beam is modeled by $y=2x$. Where do they intersect? (Solve the system; intersection points are the solutions.) $\begin{cases} y = -x^2 + 6x \\ y = 2x \end{cases}$
19. Convert the complex number $-\sqrt{3}+i$ from rectangular form to polar form $r(\cos\theta+i\sin\theta)$ with $\theta$ in degrees. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\left(\frac{b}{a}\right)$ with quadrant adjustment.
20. Multiply and write the result in standard form. This demonstrates the closure property because the product of polynomials is also a polynomial: $\left(x^2 - 3x + 2\right) \times \left(x + 4\right).$

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