Practice Test 46 - Algebra 2

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

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1. Consider the term $6x^2y$. Which list correctly gives its factors? (Factors are parts multiplied together.)
2. Restrict $f(x)=x^2$ to the domain $x\ge 0$ so that it passes the horizontal line test. After this restriction, what is the inverse function $f^{-1}(x)$?
3. A company's number of subscribers changes as shown: Week: 0, 1, 2, 3 Subscribers: 1000, 1080, 1166.4, 1259.712 Classify the pattern and identify the constant percent rate per week.
4. The polynomial $P(x) = x^3 + ax^2 + bx + c$ satisfies $P(2) = 0$ and $P(-1) = 6$. If $Q(x) = P(x) - 3$, what is the remainder when $Q(x)$ is divided by $(x + 1)$?
5. A polynomial function $f(x)$ has the factored form $f(x) = a(x+4)(x-1)^2(x-6)$ where $a$ is a nonzero constant. If the y-intercept of the graph is 48, what is the value of $a$, and how does this affect the graph's behavior at $x = 1$?
6. To factor $x^4 - 13x^2 + 36$ most efficiently, which structural approach should be used first?
7. Which matrix equation correctly represents the system (use variable order $x, y$)? $$\begin{cases} -2x+5y=11\\ 4x+y=-3 \end{cases}$$
8. What is the remainder when dividing $p(x)=2x^4-x^2+5$ by $(x+1)$? Use the Remainder Theorem (remainder $=p(a)$ for divisor $(x-a)$).
9. The function $f(x) = \frac{x-3}{x^2-9}$ models the efficiency of a chemical reaction. Examining the graph, what domain restriction is necessary for this function to be meaningful in the chemical context?
10. Restrict the domain of $f(x)=(x-1)^2$ to $x\ge 1$ so that $f$ is invertible. What is $f^{-1}(x)$?
11. Expand $(x-2)^4$ using the Binomial Theorem (Pascal's Triangle row 4: $1,4,6,4,1$).​
12. Derive the finite geometric series sum formula. Start with $S_n=a+ar+ar^2+\cdots+ar^{n-1}$ for $r\ne 1$. Multiply both sides by $r$, subtract to show the middle terms cancel (telescoping), and solve for $S_n$. Which expression is correct?​
13. A population model is defined by $P_0 = 100$ and $P_n = 1.2P_{n-1} - 5$ for $n \geq 1$. If we interpret this as a function $g(n) = P_n$, what is $g(3) - g(1)$?
14. An investment account starts with $1200$ and earns $6\%$ interest compounded annually. Model the situation with an exponential equation and solve for how many years $t$ it will take the balance to reach $2000$. Let $t$ = time in years.
15. If $i^{4k+2} = -1$ where $k$ is an integer, and $z = 3 + 4i^{4k+1}$, what is $z$ in standard form?
16. A rectangular garden has a perimeter of 40 feet. Let $L$ be the length (feet) and $W$ be the width (feet). 1) Write an equation relating $L$ and $W$. 2) Choose appropriate axes labels and scale to graph the relationship in the first quadrant (where $L\ge 0$ and $W\ge 0$). Which choice is correct?​
17. A geometric series is the sum of terms of a geometric sequence. Find the sum of the first $6$ terms of the geometric series $3+6+12+24+48+96.$ (Here $a=3$, $r=2$, $n=6$.)
18. Plotting complex numbers uses the point $(a,b)$ to represent $a+bi$ on the complex plane. If $z=2+i$, what complex number corresponds to the point obtained by multiplying by $i$ (i.e., $iz$), and what geometric transformation does this represent?
19. For $q(x)=2(x-3)(x+1)(x+2),$ identify the zeros and state the end behavior.
20. Rewrite $\frac{x^2+5x+7}{x+2}$ in the form $q(x)+\frac{r(x)}{x+2}$, where $\deg(r)<\deg(x+2)$. (Inspection is allowed.)

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