Practice Test 85 - Algebra 2

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

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1. Divide and express the result in quotient-plus-remainder form: $\frac{x^3-8}{x-2}=q(x)+\frac{r(x)}{x-2},\quad \deg(r)<1.$
2. Solve the system using the substitution method: $\begin{cases} x - y = 1\\ y = 3x - 5 \end{cases}$ What is the solution $(x, y)$?
3. Add and simplify to lowest terms. (Rational expressions behave like fractions: find an LCD, combine, and reduce.) $\frac{3}{x-2}+\frac{2}{x+2}$ (Assume $x\neq 2,-2$.)
4. A car's value is recorded at the end of each year. Year: 0, 1, 2, 3 Value ($): 18000, 15300, 13005, 11054.25 Classify the pattern as exponential growth, exponential decay, or neither, and identify the percent rate per year.
5. Use structure to factor the expression completely. Notice that it can be seen as a difference of squares more than once: $x^4 - y^4$
6. Determine whether $f(x)=x^4-2x^2$ is even, odd, or neither.​
7. Find the modulus $r$ and an argument $\theta$ (in radians) for the complex number $-\sqrt{3}-i$, then choose its polar form $r(\cos\theta+i\sin\theta)$. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\left(\frac{b}{a}\right)$ with quadrant adjustment. (Argument is measured counterclockwise from the positive real axis.)
8. Write an expression for $f^{-1}(x)$ given $f(x)=3x-7$.​​​
9. A sequence can be viewed as a discrete function $f$ whose domain is the positive integers $n \in \{1,2,3,\dots\}$. For the Fibonacci sequence defined by $f(1)=f(2)=1$ and $f(n)=f(n-1)+f(n-2)$ for $n\ge 3$, what is $f(7)$?​
10. What is the coefficient of $x^2y^3$ in the expansion of $(x+y)^5?$ (You may use Pascal's Triangle.)​
11. A one-to-one function $f$ is shown by the table, so $f$ has an inverse. Use the table to evaluate $f^{-1}(21)$. (Because inverse pairs swap, if $f(x)=21$ then $f^{-1}(21)=x$.)

    $x$	$f(x)$
    2	9
    4	13
    6	17
    8	21
    10	25

12. Apply the sum of cubes identity to factor the polynomial $x^3+8$ completely over the integers.​
13. The function $f(x)=\sqrt{x-2}$ can be used to model the side length (in meters) of a square when $x$ represents the measured area (in square meters) plus 2. What is the realistic domain for $x$?​
14. Use technology to solve $AX=B$ by computing $X=A^{-1}B$. $A=\begin{bmatrix}2&1&0\\1&3&1\\0&2&4\end{bmatrix},\quad B=\begin{bmatrix}5\\10\\16\end{bmatrix}$ What is $X$?
15. A physics model for distance is $d(t)=v\,(t-2)^2+k,$ where $v$ and $k$ are constants and $t$ is time. By chunking $(t-2)^2$ as a single entity, which statement correctly identifies the part independent of $v$?
16. The expression $\sqrt{x+1} \cdot (x+1)^{3/2} + 5\sqrt{x+1} - 2(x+1)^{5/2}$ involves multiple powers of the same radical. Which substitution best reveals the polynomial structure?
17. For the system below (variable order $x, y, z$), identify the coefficient matrix $A$ in $AX=B$: $$\begin{cases} -x+4y+0z=2\\ 3x-y+5z=-1\\ 2x+0y-z=7 \end{cases}$$
18. Solve the system algebraically. The solution(s) are the intersection point(s) of the line and the parabola. $\begin{cases} y = -x + 1 \\ y = x^2 + 2x - 3 \end{cases}$ How many real solutions does the system have?​
19. Use the inverse property $\log_b(b^x)=x$ to simplify $\log_3\left(3^8\right)$.
20. A projectile's horizontal distance is $x(t) = 40t$ and vertical distance is $y(t) = 60t - 16t^2$ feet after $t$ seconds. The trajectory function $h(x) = 1.5x - 0.01x^2$ gives height as a function of horizontal distance. What domain for $h(x)$ corresponds to the projectile's flight?

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