Practice Test 52 - Algebra 2

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

A polynomial identity such as $(a+b)^2=a^2+2ab+b^2$ is known to hold for real numbers. Which option correctly verifies that it also holds for complex numbers by using $a=1+i$ and $b=2-i$ ?
Divide and simplify to lowest terms by multiplying by the reciprocal ("keep, change, flip"). This demonstrates closure under division by a nonzero rational expression. Divide: $\displaystyle \frac{x^2+5x+6}{x^2-1}\div\frac{x+2}{x-1}$
Solve using the quadratic formula: $2x^2 + 3x - 7 = 0$ .
The value of a car depreciates according to $V(t) = 25000(0.82)^t$ , where $t$ is years since purchase. The owner also considers that inflation affects purchasing power at 3% annually. Relative to inflation, how does the car's value change each year?
The function $f(x)=x^2$ (domain: all real numbers) is not invertible because it fails the horizontal line test. Which domain restriction makes $f$ one-to-one (and therefore invertible)?
Evaluate the natural logarithm $\ln(e^5)$ using the inverse relationship between $e^x$ and $\ln(x)$ .
A radioactive substance decays according to $N(t) = N_0 \cdot 16^{-t/8}$ , where $t$ is in years. Which transformation reveals the half-life of the substance?
In the expression $4x^2 - 3x + 9$ , how many terms are there, and what are they? (Remember: terms are separated by $+$ or $-$ , and the sign belongs to the term.)
In the expression $4x^2 + 3x$ , is the number $4$ best described as a term, factor, or coefficient?
Verify the claim using the Remainder Theorem: When dividing $p(x)=x^4-3x^2+2x+1$ by $(x-1)$ , the remainder is $1$ .
A school club is ordering T-shirts and hoodies for a fundraiser. Let $x$ = number of T-shirts and $y$ = number of hoodies. Constraints: - Each T-shirt costs $8 and each hoodie costs $20; the club has at most $400 to spend. - They need at least 30 total items. - They must order at least 10 hoodies. - They cannot order negative amounts. Which system of inequalities represents all constraints?
Determine the inverse function for $f(x)=x^3+5$ .
Consider the polynomial $R(x) = 3x^3 - 7x^2 + 4x - 12$ . If $(x - k)$ is a factor of $R(x)$ , which of the following could be the remainder when $R(x)$ is divided by $(x - 2k)$ ?
Solve the $2\times 2$ system using the inverse-matrix method. First find the inverse of the coefficient matrix, then compute $X=A^{-1}B$ . $\begin{cases} 2x+y=5\\ 3x+4y=7 \end{cases}$ What is $X=\begin{bmatrix}x\\y\end{bmatrix}$ ?
Verify the geometric series formula by computing both ways: Evaluate $S_4=3+6+12+24$ using $S_n=\dfrac{a(1-r^n)}{1-r}$ with $a=3$ , $r=2$ , $n=4$ .
Let the parent function be $f(x)=x^2$ . What transformation changes $f(x)$ to $g(x)=f(x+2)$ ?
A sequence is a function whose domain is a subset of the integers, so it is defined only at discrete inputs. Which set is the most appropriate domain for a sequence written as $a_1, a_2, a_3, \dots$ ?
Let $E = 5(x-2)^2 + 7(x-2) - 3.$ To see the structure within this complicated expression, you are told to chunk by letting $u = (x-2)$ . Which rewritten form best shows $E$ as a polynomial in the single entity $u$ ?
A quadratic expression $ax^2 + bx + c$ has zeros at $x = -\frac{2}{3}$ and $x = \frac{1}{4}$ . If $a = 12$ , what is the value of $c$ ?
A researcher observes that a bacterial culture triples every 4 hours. If the initial population is 500 bacteria, and she wants to model this with the form $N(t) = 500 \cdot b^t$ where $t$ is in hours, what is the hourly percent growth rate of this culture?

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