Practice Test 83 - Algebra 2

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

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1. In the rational expression $\dfrac{3x^2 - 2x + 5}{x-1}$, identify the terms in the numerator.
2. The system of equations below is to be written in matrix form $A\vec{x} = \vec{b}$: $x - 3z = 5$ $2y + 4z = -1$ $3x + y - z = 7$ What is the sum of all entries in the first column of matrix $A$?
3. Identify the vertical asymptote(s) and any holes of the rational function $f(x) = \frac{(x-4)(x+1)}{(x-4)(x-2)}.$
4. Rewrite $\frac{x^2+5x+7}{x+2}$ in the form $q(x)+\frac{r(x)}{x+2}$ (inspection is fine). Ensure $\deg(r)<\deg(x+2)$.
5. For $ax^2+bx+c=0$, the discriminant $b^2-4ac$ determines the solution type (2 distinct real, 1 repeated real with multiplicity 2, or 2 complex conjugates), but every quadratic has exactly 2 complex solutions counting multiplicity. For $x^2+2x+5=0$, which choice correctly describes the solutions?
6. Verify or disprove: $f(x)=\dfrac{x-2}{3}$ and $g(x)=3x-2$ are inverses. Use composition to compute $f(g(x))$ and $g(f(x))$ and check whether both equal $x$.
7. A lab culture is counted once per hour: Hour: 0, 1, 2, 3 Cells: 5,000; 5,750; 6,612.5; 7,604.375 Does this represent constant percent change? If yes, what is the percent growth rate per hour?
8. Complete the square to solve and express any nonreal solutions in $a+bi$ form: $x^2-6x+13=0.$
9. Use the identity $(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2$ to generate a Pythagorean triple when $x=3$ and $y=2$. (A Pythagorean triple is $(a,b,c)$ with $a^2 + b^2 = c^2$.)
10. Two complex numbers $z_1 = 2 + 3i$ and $z_2 = 1 - 2i$ are added geometrically on the complex plane. If the parallelogram method is used to represent this addition, what are the coordinates of the vertex opposite to the origin?
11. A parabola crosses the x-axis where $g(x)=x^2-25$ equals zero. Factor $g(x)$ and identify the zeros and x-intercepts.
12. A gym charges a $25 monthly membership fee plus $5 per class attended. Let $M$ be the total monthly cost (dollars) and $c$ be the number of classes. What equation models the situation, and what is a reasonable y-axis scale to graph the cost for $0\le c\le 15$?
13. A ball is thrown upward from a 6-foot platform. Its height is modeled by $h(t) = -16t^2 + 32t + 6$ feet after $t$ seconds. The ball is caught when it returns to the platform height. What domain restriction best reflects this scenario?
14. A company's revenue is modeled by $R = p\,q,$ where $p$ is price per item and $q$ is quantity sold. If you view $q$ as a single entity independent of $p$, which interpretation is correct?
15. Graph $p(x)=x(x-2)^2(x+1)$ showing all zeros, indicating which zeros the graph touches or crosses, and giving the end behavior.
16. Consider the set of all complex numbers $z=a+bi$ such that $|z|=5$, where $|z|=\sqrt{a^2+b^2}$. What does this set represent geometrically on the complex plane?
17. Complete the square by recognizing perfect square structure and rewrite: $x^2 + 8x + 20$
18. Compute the product $(4 + 3i)(4 - 3i)$. (Use the key property $(a+bi)(a-bi)=a^2+b^2$, which is always real.)
19. For the exponential function $g(x)=2\cdot 3^x$, verify using equal intervals of length $1$ that the function grows by equal factors. Compute $g(0),g(1),g(2),g(3)$ and the successive ratios, and identify the correct conclusion.
20. The function $f(x)=-x^2+10$ (domain: all real numbers) fails the horizontal line test. Which domain restriction makes $f$ one-to-one and invertible?

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