Practice Test 91 - Algebra 2

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

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1. The identity $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ works for complex $a, b$ as well. Let $a = i$ and $b = 1$. What is the value of $a^3 + b^3$?
2. Use zeros to construct a rough graph of the polynomial $p(x)=(x+2)(x-1)^2(x-4).$ Identify all zeros with multiplicities, state whether the graph crosses or touches the x-axis at each zero, and determine the end behavior.
3. A company models its weekly profit (in thousands of dollars) by the function $P(x)=-x^2+6x-5$, where $x$ is the number of weeks since a new product launch. Interpret the x-intercepts of $P$ in this context by identifying them as points.
4. A researcher observes that for $0 ≤ x ≤ 8$, the function $p(x) = x^3 - 2x^2 + 50$ consistently produces larger values than $q(x) = 3 \cdot 1.4^x$. She concludes that cubic functions grow faster than exponential functions. What is the fundamental flaw in this reasoning?
5. Solve using the quadratic formula: $2x^2-3x-7=0$.
6. For $p(x)=-(x+4)^2(x-1)^3$, sketch $p(x)$ showing zeros, multiplicities, and end behavior. Which statement is correct?
7. Consider the polynomial identity $x^4 + x^2y^2 + y^4 = (x^2 + y^2)^2 - x^2y^2$. If this identity is applied to simplify $16t^4 + 4t^2s^2 + s^4$, what is the resulting expression in terms of perfect squares?
8. Use a calculator to solve the exponential equation $6\cdot 10^{4t}=95$. Give $t$ in exact logarithmic form and as an approximation.
9. For the function $p(x) = \frac{x + 5}{3x - 2}$ where $x \neq \frac{2}{3}$, what value of $x$ satisfies $p(x) = \frac{1}{2}$?
10. Let $z = 7 - 4i$. Which expression equals the complex conjugate $\overline{z}$ (recall: the conjugate flips the sign of the imaginary part only)?
11. A population model uses the annual growth expression $(1.08)^t$, where $t$ is in years. Which equivalent expression best reveals the quarterly growth factor (4 quarters per year)?​​
12. In the expression $3(x+2)^2 + 5x - 1$, identify all the terms.
13. An electronics store's weekly revenue is modeled by $R = 150n + 2000$, where $R$ is revenue in dollars and $n$ is the number of laptops sold that week. In this model, what does the parameter $2000$ represent?
14. An exponential function has the form $f(x)=a\cdot b^x$ and passes through $(1,16)$ and $(3,64)$. Which function is correct?
15. Use the logarithm properties $\log_b(xy)=\log_b(x)+\log_b(y)$, $\log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y)$, and $\log_b(x^p)=p\log_b(x)$ to expand: $\ln\left(\frac{a^4b}{\sqrt{c}}\right).$
16. Solve the quadratic equation by factoring: $x^2 - 9x + 20 = 0$.
17. Let $z=3+4i$ and $w=1-i$. On the complex plane, the distance between the points representing $z$ and $w$ is $|z-w|$. What is this distance?
18. A taxi fare is based on a $3.50 starting fee plus $2.25 per mile. Write an equation for the fare $F$ (dollars) in terms of miles $m$, and choose appropriate axes labels and a reasonable scale to graph the fare for $0 \le m \le 15$.
19. A chemical solution has pH 6.3. Using $\text{pH}=-\log_{10}([\text{H}^+])$, determine the hydrogen ion concentration $[\text{H}^+]$ in moles per liter. Round to two significant figures.
20. When simplifying $\frac{x^2 - 9}{x^2 + 6x + 9} \cdot \frac{x + 3}{x - 3}$, what restrictions must be placed on the domain?

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