Practice Test 104 - 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 expression $E = 3x(2x-5)^2.$ View $(2x-5)$ as a single entity (chunk). Which statement correctly describes how $E$ is built from independent parts?
2. Find $x$ in the equation $7\cdot 10^{x}=320$. Express the solution as a logarithm (exact form) and then approximate using a calculator.
3. Evaluate $\ln(e^6)$ using the inverse relationship between $e^x$ and $\ln(x)$.
4. For the equation $x^2 - 6x + 10 = 0$, the discriminant is negative. Solve and write the solutions in $a \pm bi$ form.
5. Which expression is equivalent to $2\log_5(x) - \frac{1}{2}\log_5(y) + 3\log_5(z)$?
6. In the expression $-x + 7$, what is the coefficient of $x$? (Recall that $-x$ means $-1\cdot x$.)
7. Synthetic division can evaluate a polynomial efficiently: when dividing by $(x-c)$, the remainder equals $P(c)$ (Remainder Theorem). Use synthetic division with $c=2$ to evaluate $P(x)=x^3-x^2-4x+4$ by finding the remainder when $P(x)$ is divided by $(x-2)$. What is $P(2)$?
8. Compute the finite geometric sum $S_8=7+7\left(\tfrac12\right)+7\left(\tfrac12\right)^2+\cdots+7\left(\tfrac12\right)^7.$ (Note: This is finite; if $|r|<1$, an infinite sum would converge, but here you are finding $S_8$.)
9. Use polynomial division to rewrite $\frac{x^3 - 8}{x - 2}$ in the form $q(x) + \frac{r(x)}{x-2}$, where $\deg(r) < 1$.
10. Use polynomial division (inspection or long division) to rewrite the rational expression $\frac{x^3+2x^2-5x+3}{x-1}$ in the form $q(x)+\frac{r(x)}{x-1}$, where $\deg(r)<\deg(x-1)$.
11. Which equation is factoring most efficient for solving? A. $x^2+5x+6=0$ B. $x^2+5x+1=0$ C. $2x^2+3x-7=0$ D. $x^2+4x+8=0$
12. Sketch the polynomial $f(x)=(x-2)(x+1)(x+4)$ identifying all zeros and the end behavior. Which description matches the correct sketch?​
13. Multiply the polynomials (use the distributive property). This demonstrates closure because the product of polynomials is a polynomial: $(x^2-3x+2)\times(x+4).$
14. Sketch a graph of a function $f(x)$ that models a hiker's elevation change (in hundreds of meters) versus time $x$ (in hours) and has all of these features: - $y$-intercept at $(0,1)$ - x-intercepts at $(2,0)$ and $(6,0)$ - Increasing on $(0,4)$ and decreasing on $(4,8)$ - A relative (and absolute) maximum at $(4,3)$ - End behavior: as $x\to\infty$, $f(x)\to-\infty$ and as $x\to-\infty$, $f(x)\to-\infty$ Which sketch matches these features?
15. A student is mixing two types of paint to make custom posters. Let $x$ be liters of Paint A and $y$ be liters of Paint B. Constraints: - Total paint needed is at least 5 liters: $x+y\ge 5$. - Paint A is limited to at most 3 liters: $x\le 3$. - The mixture must have at least as much Paint B as Paint A: $y\ge x$. - Non-negativity: $x\ge 0$, $y\ge 0$. Which point is in the feasible region (satisfies all constraints)?
16. Given that $u_n = 6 \cdot (\frac{1}{2})^{n-1}$ for $n \geq 1$, which recursive formula is equivalent to this explicit formula, and what is the value of $u_5$?
17. A polynomial $f(x)$ has degree 4, and when divided by $(x + 3)$ using synthetic division, the bottom row of the synthetic division table shows: $1, -2, 5, -1, 4$. Which statement about $f(x)$ is correct?
18. The quadratic $3x^2 + 12x + 15 = 0$ has solutions of the form $a \pm bi$. What are the values of $a$ and $b$?
19. Use the geometric series sum formula $S_n = \dfrac{a(1-r^n)}{1-r}$ (for $r\ne 1$) to find the value of $2 + 2(1.1) + 2(1.1)^2 + \cdots + 2(1.1)^9.$
20. Verify algebraically that $g(x)=4\cdot(1.5)^x$ has a constant ratio over equal intervals of length $1$ by computing $\dfrac{g(x+1)}{g(x)}$. What is the ratio, and what does it imply about the sampled values $g(0),g(1),g(2),\dots$?

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