Practice Test 21 - Algebra 2

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

Use the Rational Zeros Theorem to list all possible rational zeros of $P(x)=3x^3-2x^2+x-6.$ (Remember: any rational zero $\frac{p}{q}$ in lowest terms has $p\mid$ constant term and $q\mid$ leading coefficient.)
Demonstrate that the polynomial identity $(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$ holds for all real $x,y$ by expanding both sides and simplifying.
Four functions are defined for $x\ge 0$ : linear $f(x)=20x$ , quadratic $g(x)=5x^2$ , polynomial $h(x)=x^3$ , and exponential $p(x)=2^x$ . Which statement is correct about their long-term behavior?
Compute the product and note it is always real and nonnegative: $(6-2i)(6+2i).$
Condense to a single logarithm: $\log_5(x)-2\log_5(y)+\log_5(z)$ (Assume $x>0$ , $y>0$ , and $z>0$ .)
A savings account balance (in dollars) after $t$ years is modeled by $B(t)=500(1.03)^t$ . Mathematically, the formula works for all real $t$ . Realistically, the account has been open for at most 10 years and $t$ measures time since opening. Which choice correctly compares the mathematical and realistic domains?
Function $f$ is given by $f(x)=|x-3|+2$ . Function $g$ is described verbally as: " $g$ is an absolute value function with minimum value 1 occurring at $x=-2$ ." Which function has the smaller minimum value?
A system is written in matrix form $AX=B$ . Determine whether $A$ is invertible (has an inverse). If it is invertible, you may solve using $X=A^{-1}B$ . $A=\begin{bmatrix}1&2&3\\2&4&6\\1&0&1\end{bmatrix},\quad X=\begin{bmatrix}x\\y\\z\end{bmatrix},\quad B=\begin{bmatrix}1\\2\\3\end{bmatrix}.$ Which statement is true?
Find the modulus $r$ and argument $\theta$ (in degrees) of the complex number $3 - 3i$ . Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\left(\frac{b}{a}\right)$ with quadrant adjustment (argument is the angle from the positive real axis). Which polar form is correct?
Consider the rational function $f(x) = \frac{x^2 - 4x - 5}{x^2 - 9}$ . After factoring completely, which statement correctly describes the behavior of this function?
Show that any exponential function $g(x)=a\cdot b^x$ (with $a\neq 0$ , $b>0$ , $b\neq 1$ ) grows by equal factors over equal intervals. Specifically, prove that for any real number $h$ , the ratio $\dfrac{g(x+h)}{g(x)}$ is constant with respect to $x$ , and connect this to why $\{g(n)\}$ at integer inputs is a geometric sequence.
For the logarithmic function $h(x) = \ln(x-3)$ , what is the vertical asymptote?
On the complex plane, let $z_1=3+2i$ and $z_2=1-i$ . The distance between two complex numbers is $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$ , and the midpoint is $\dfrac{z_1+z_2}{2}=\left(\dfrac{a+c}{2}\right)+\left(\dfrac{b+d}{2}\right)i$ . Which option gives both the distance from $z_1$ to $z_2$ and the midpoint of the segment connecting them (same formulas as in coordinate geometry for points $(a,b)$ and $(c,d)$ )?
Find the quotient $q(x)$ and remainder $r(x)$ when dividing $x^3 + 4x^2 - 2x + 7$ by $x + 2$ , and write $\frac{x^3 + 4x^2 - 2x + 7}{x + 2} = q(x) + \frac{r(x)}{x + 2}$ with $\deg(r) < \deg(x + 2)$ .
On the complex plane, $z = 6 - 8i$ corresponds to the point $(6,-8)$ . Using $|a+bi|=\sqrt{a^2+b^2}$ , what is $|z|$ ?
Identify the zeros and their multiplicities for $p(x)=x^2(x-4)(x+1)^2$ and determine whether the graph crosses or touches at each zero.
Consider the functions $f(x)=5x$ (linear), $g(x)=2x^2$ (quadratic), $h(x)=x^4$ (polynomial), and $p(x)=1.5^x$ (exponential). Which function eventually grows fastest as $x\to\infty$ ?
An investment grows according to $A(t)=P\,(1.15)^t$ (with $t$ in years). Which statement correctly interprets the transformation $(1.15)^t = \left(1.15^{1/12}\right)^{12t}?$
A water tank is being drained at a rate that depends on the height of water remaining. The relationship between time $t$ (in minutes) and height $h$ (in feet) is inverse: as time increases, height decreases such that their product remains constant at 240. Which equation and graph characteristics correctly describe this situation?
Functions $s$ and $t$ are inverses of each other. If $s(3) = 7$ and $s(5) = 2$ , what is the value of $t(7) - t(2)$ ?

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