Practice Test 26 - Algebra 2

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

Divide $x^4+x^3-2x+3$ by $x^2-1$ and write $\frac{x^4+x^3-2x+3}{x^2-1}$ as $q(x)+\frac{r(x)}{x^2-1}$ , where $\deg(r)<2$ .
The pH of a solution is given by $\text{pH}=-\log_{10}([\text{H}^+])$ , where $[\text{H}^+]$ is the hydrogen ion concentration in moles per liter. A solution has $[\text{H}^+]=3.2\times 10^{-5}$ . What is the pH? Round to the nearest hundredth.
Solve the system and give the exact solution (an ordered pair). $\begin{cases} y = 3x + 1 \\ y = -2x + 11 \end{cases}$
On the complex plane, $z=a+bi$ corresponds to $(a,b)$ . The distance between $z_1=a+bi$ and $z_2=c+di$ is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$ , and the midpoint is $\dfrac{z_1+z_2}{2}=\dfrac{a+c}{2}+\dfrac{b+d}{2}i$ . Let $z_1=3+2i$ and $z_2=-1+2i$ . What are the distance and the midpoint?
Solve the system using the inverse of the coefficient matrix. \begin{align*} 2x+y&=5\\ 3x+4y&=7 \end{align*} Let $A=\begin{bmatrix}2&1\\3&4\end{bmatrix}$ , $X=\begin{bmatrix}x\\y\end{bmatrix}$ , and $B=\begin{bmatrix}5\\7\end{bmatrix}$ . Compute $A^{-1}$ (by hand) and then $X=A^{-1}B$ . What is $(x,y)$ ?
A medication dose begins at 120 mg in the bloodstream. Each hour, $12\%$ of the medication is eliminated. Create an exponential decay model for the amount $M(t)$ remaining after $t$ hours, then determine how long it takes for the amount to drop to 30 mg. Round to the nearest tenth of an hour and interpret your result in context.
After restricting $f(x)=(x-1)^2$ to the domain $x\ge 1$ so it passes the horizontal line test, what is $f^{-1}(x)$ ?
Let $z=2+5i$ and $w=1-3i$ . Find $\overline{(z+w)}$ . (Use the property $\overline{z+w}=\overline{z}+\overline{w}$ .)
In optics, thin lens relationships can be written as $\dfrac{1}{f} = \dfrac{1}{a} + \dfrac{1}{b}$ , where $f$ is focal length and $a,b$ are distances. Rearrange $\dfrac{1}{f} = \dfrac{1}{a} + \dfrac{1}{b}$ to solve for $f$ .
A bacterial culture starts with 200 bacteria and triples every 4 hours. If $b_n$ represents the number of bacteria after $4n$ hours, which explicit formula represents this growth, and approximately how many bacteria will there be after 16 hours?
Find the intersection point(s) of the line and the circle by solving the system. $\begin{cases} x^2 + y^2 = 25 \\ y = -3x \end{cases}$ What are all solutions $(x,y)$ ?
A tank is filled at a steady rate. The volume of water $V$ (in liters) after $t$ minutes is shown. $t$ : 0, 1, 2, 3, 4 $V$ : 10, 14, 18, 22, 26 Is the rate of change constant? If so, what is the constant rate $\Delta V/\Delta t$ ?
For $f(x)=\frac{x^2-9}{x+2},$ identify the zero(s), vertical asymptote(s), and oblique asymptote.
Synthetic division can evaluate a polynomial quickly because the remainder when dividing by $(x-c)$ equals $P(c)$ (Remainder Theorem). Use synthetic division on $P(x)=x^3-6x^2+11x-6$ with divisor $(x-2)$ (so $c=2$ ). What is the remainder (i.e., $P(2)$ ), and is $(x-2)$ a factor?
Find the quotient $q(x)$ and remainder $r(x)$ when dividing $a(x)=x^3+4x^2-2x+7$ by $b(x)=x+2$ , and write $\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$ with $\deg(r)<\deg(b)$ .
Apply the difference of cubes identity to factor $x^3-27$ completely over the integers.
Which of the following represents the extension of the polynomial identity $(a^2 + b^2) = (a + bi)(a - bi)$ when applied to the expression $x^4 + x^2 + 1$ ?
Find the center and radius of the circle $x^2+y^2+6x-4y-12=0$ by completing the square.
Check whether $f(x)=\dfrac{x+1}{x-2}$ and $g(x)=\dfrac{2x+1}{x-1}$ are inverse functions by verifying both compositions $f(g(x))$ and $g(f(x))$ (for values where the expressions are defined).
A rectangular patio has an area of 96 square feet. Its length is 4 feet more than its width. Write and solve an equation to find the width of the patio. Let $w$ = the width (in feet).

Practice tests