Practice Test 84 - Algebra 2

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

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1. Use polynomial division to rewrite $\frac{2x^3+x^2-5x+3}{x-2}$ in the form $q(x)+\frac{r(x)}{x-2}$, where $\deg(r)<1$.
2. Solve $2\cdot 10^{3x}=65$ for $x$. Express the solution as a logarithm and evaluate using technology.
3. Let $z=3+2i$. The conjugate of $z$ is $\overline{z}=3-2i$. On the complex plane (real axis horizontal, imaginary axis vertical), how are $z$ and $\overline{z}$ related geometrically?
4. In physics, the kinematic equation $v^2 = u^2 + 2as$ relates final speed $v$, initial speed $u$, acceleration $a$, and displacement $s$. Solve $v^2 = u^2 + 2as$ for $a$ (treating the other variables as constants, as in solving a numeric equation).
5. Use the inverse relationship to evaluate $10^{\log(25)}$, where $\log(x)$ means $\log_{10}(x)$.
6. A electronics manufacturer produces smartphones and tablets. Due to component shortages, they can make at most $300$ devices per week. Smartphones require $2$ hours of assembly time each, while tablets require $3$ hours each. The company has $750$ hours of assembly time available weekly. Market demand requires producing at least twice as many smartphones as tablets. Which inequality represents the market demand constraint if $s$ is smartphones and $t$ is tablets?
7. The pH of a solution is given by $pH = -\log[H^+]$, where $[H^+]$ is the hydrogen ion concentration in moles per liter. If the pH of a solution increases from 3 to 5, by what factor did the hydrogen ion concentration change?
8. What are the center and radius of the circle given by $ $(x+5)^2$ + $(y-1)^2$ = 49 $?
9. A bacteria culture starts with 200 bacteria and grows by 10% each hour. Let $B(t)$ be the number of bacteria after $t$ hours. Which exponential function models this situation?
10. Solve the system by elimination. Give the solution as an ordered pair $(x,y)$. $\begin{cases} x + y = 7 \\ 2x - y = 8 \end{cases}$
11. An investment of $1200 earns 6% interest compounded annually. Model the balance with an exponential equation and solve for how many years it takes for the investment to reach $2000. Let $t$ = the number of years.
12. Over the real numbers, a quadratic can have 0, 1, or 2 real solutions depending on the discriminant. Over the complex numbers, a quadratic $ax^2+bx+c=0$ always has exactly 2 solutions counting multiplicity (either 2 distinct real, 1 repeated real counted twice, or 2 complex conjugates). Which statement is correct?
13. Based on the graph shown, which system of equations best represents the two lines?
14. Use Pascal's Triangle (row 4: $1,4,6,4,1$) to expand $(x-y)^4.$
15. Sketch $p(x)=-x(x-3)^2(x+2)$ showing zeros with multiplicities, crossing vs touching, and end behavior.
16. Write the system as $AX=B$ (variable order $x, y, z$). Be sure to include $0$ for any missing variable: $\begin{cases} x+3y=5\\ 2x-z=1\\ y+2z=4 \end{cases}$
17. Four functions are defined: linear $f(x)=x$, quadratic $g(x)=x^2$, polynomial $h(x)=x^6$, and exponential $p(x)=2^x$. Which function eventually grows fastest as $x\to\infty$?
18. Use the conjugate property to simplify. What is $(2+5i)(2-5i)?$
19. Use the Rational Zeros Theorem to help factor completely: $P(x)=x^3-7x-6.$ First list possible rational zeros, then test to find an actual zero, and factor the polynomial.
20. Find the inverse function $f^{-1}(x)$ for $f(x)=\dfrac{4}{x+1}$ with domain restriction $x\ne -1$.​

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