Practice Test 54 - Algebra 2

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

A polynomial models the net revenue (in thousands of dollars) from ticket sales for an event: $R(x)=(x+1)(x-4)(x-6),$ where $x$ is the ticket price (in dollars). On what interval is $R(x)$ positive?
In the complex plane, consider the quadrilateral with vertices at $2 + i$ , $4 + 3i$ , $2 + 5i$ , and $0 + 3i$ . What is the length of the diagonal connecting $2 + i$ and $2 + 5i$ ?
Consider the polynomial $P(x) = 2x^3 - x^2(x + 4) + 3x - 1$ . When $P(x)$ is written in standard form, what is the sum of all the coefficients?
A landscaping company models the length of fencing needed (in meters) for a square garden with area $x$ square meters by $F(x)=4\sqrt{x}$ . In context, area cannot be negative. What is the realistic domain for $x$ ?
Evaluate $\ln(e^4)$ using the inverse relationship between $e^x$ and $\ln(x)$ .
A student claims $f(x)=x^2$ (with domain restricted to $x\ge 0$ ) and $g(x)=\sqrt{x}$ are inverses. Verify the claim by showing whether both $f(g(x))$ and $g(f(x))$ equal $x$ on the appropriate domains.
Complete the square to rewrite $x^2+y^2+10x-2y+17=0$ in standard form $(x-h)^2+(y-k)^2=r^2$ .
A population model is given by $f(x)=-2x^3+5x$ , where $x$ is time in years. Describe the end behavior of $f$ .
A school is ordering $n$ identical calculators for students. The total cost is $C(n)=12.50n$ . Should the graph of $C$ versus $n$ be shown as discrete points or as a continuous curve, and why?
Use logarithms to solve the exponential equation $3\cdot 10^{x}=58$ for $x$ . Give the exact answer (as a logarithm) and an approximate value using technology.
Apply the difference of cubes identity to factor the polynomial completely: $x^3 - 8$ .
Consider the expression $E = 3x\,(x^2+2x-5).$ If you view $(x^2+2x-5)$ as a single unit, which statement correctly describes how $E$ changes when only the coefficient $3$ is replaced by $6$ (with $x$ unchanged)?
Use the difference of squares identity to factor $x^2+4$ completely over $\mathbb{C}$ (even though it does not factor over the reals).
An investment grows according to $A(t)=P\,(1.15)^t$ (with $t$ in years). Which statement correctly interprets the transformation $(1.15)^t = \left(1.15^{1/12}\right)^{12t}?$
Demonstrate that the identity $a^2-b^2=(a+b)(a-b)$ holds for all real $a,b$ by expanding the right-hand side. Which expansion is correct?
Sketch $f(x)=-2(x+3)^2(x-1)(x-4)$ showing zeros, their multiplicities, and end behavior. Which description matches the correct rough graph?
Use logarithms to solve the exponential equation $3\cdot 10^{2t}=75$ for $t$ . Give the exact solution (as a logarithm) and an approximate value using technology.
Use polynomial division to write $\frac{2x^4 + 3x^3 - x + 5}{x^2 + 1}$ in the form $q(x) + \frac{r(x)}{x^2+1}$ , where $\deg(r) < 2$ .
What is the LCD (least common denominator) for the rational expressions $\frac{3}{x^2-4} \quad\text{and}\quad \frac{2}{x^2+4x+4}?$ (Factor denominators first.)
The function $f(x)=x^2$ (domain: all real numbers) is not invertible because it fails the horizontal line test (for example, $f(2)=f(-2)=4$ ). Which domain restriction makes $f$ one-to-one and therefore invertible?

Practice tests