Practice Test 22 - Algebra 2

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

On the same axes, $f(x)=2^x$ and $g(x)=\log_2(x)$ are graphed. Which statement best describes their relationship?
For $z=-6+8i$ , compute $z\overline{z}$ . (Recall $z\overline{z}=a^2+b^2=|z|^2$ , a real positive number.)
For $f(x)=\frac{(x-3)(x+2)}{(x-3)(x-1)},$ identify the zero(s), vertical asymptote(s), and any hole (removable discontinuity).
A landscaper models the length of a walkway (in feet) that can be built from $x$ paving stones using $L(x)=2x+3$ . The number of paving stones must be a whole number, and the landscaper has at most 50 stones available. Which statement best describes how the graph of $L$ should be represented for this context?
Multiply the polynomials (use the distributive property). This demonstrates closure because the product of polynomials is a polynomial: $(x^2-3x+2) \times (x+4).$
If the remainder when $p(x)=x^3-2x^2+kx-4$ is divided by $(x-2)$ is $6$ , what is the value of $k$ ? (Use remainder $=p(2)$ .)
The complex plane below shows the real axis (horizontal) and imaginary axis (vertical). Points $P$ and $Q$ are plotted at $P(3,2)$ and $Q(-1,-4)$ . Which pair of complex numbers matches points $P$ and $Q$ ? (Recall: the point $(a,b)$ represents $a+bi$ .)
A population is modeled by $P(t)=P_0(1.08)^t$ , where $t$ is in years. Rewrite the growth part $(1.08)^t$ to show an equivalent form with a quarterly growth factor raised to the number of quarters. (Use the power-of-power property.)
Transform and solve by completing the square: $x^2+10x-3=0$ Write it as $(x-p)^2=q$ and then solve for $x$ .
Use the discriminant to determine the solution type for $x^2-5x+6=0$ . Remember: every quadratic $ax^2+bx+c=0$ has exactly 2 complex solutions counting multiplicity (2 distinct real, 1 repeated real, or 2 complex conjugates), and the discriminant $b^2-4ac$ determines which case. Which statement is correct?
A cyclist rides 20 miles at a speed of $r$ mph, then 15 miles at a speed of $(r-5)$ mph. The total riding time is 2.5 hours. Set up a rational equation and solve for the cyclist's speed $r$ on the first part. Let $r$ = speed (mph) for the first 20 miles, with $r>5$ .
Use structure to factor $x^3-27$ by recognizing a difference of cubes.
Let $f(x)=x^2-2$ and $g(x)=2x+1$ . Use the intersection points of the graphs $y=f(x)$ and $y=g(x)$ to solve $f(x)=g(x)$ . Which set of $x$ -values are the solutions?
Use synthetic division to test whether $(x+2)=(x-(-2))$ is a factor of $P(x)=x^4-2x^3-7x^2+8x+12.$ Use $c=-2$ with coefficients $1,-2,-7,8,12$ . Which statement is correct? (If the remainder is $0$ , then it is a factor.)
Divide and simplify to lowest terms. (Use "multiply by the reciprocal," like fraction division.) $\frac{x^2-1}{x^2+4x+3}\div\frac{x-1}{x+1}$ (Assume $x \neq -3, -1, 1$ .)
A quadratic polynomial with real coefficients can be written as $ax^2+bx+c$ with $a\ne 0$ . In the complex numbers it always has exactly 2 zeros counting multiplicity, which previews the Fundamental Theorem of Algebra (degree- $n$ polynomials have exactly $n$ complex zeros counting multiplicity). Which statement correctly interprets "counting multiplicity" for the quadratic $(x-3)^2=0$ ?
Represent the system as a matrix equation $AX=B$ (use variable order $x, y, z$ ): $\begin{cases} 2x+y-z=5\\ x-3y+2z=1\\ 4x+y+z=9 \end{cases}$
Consider the nested expression $J(x)=2(3x+1)^3.$ Chunk by viewing $(3x+1)$ as a single entity. Which interpretation correctly identifies what depends on $x$ and what does not?
Use synthetic division (instead of long division) to find the remainder when $P(x)=x^4-5x^2+3$ is divided by $(x+3)=(x-(-3)).$ Use $c=-3$ and remember to include a $0$ coefficient for the missing $x^3$ and $x$ terms. By the Remainder Theorem, the remainder equals $P(-3)$ . What is the remainder?
A number and its reciprocal have a sum of $\frac{10}{3}$ . Set up and solve an equation to find the number. Let $x$ = the number (and $x\ne 0$ ).

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