Practice Test 82 - Algebra 2

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

Demonstrate that the polynomial identity $(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2$ holds for all real $x$ and $y$ by expanding both sides and simplifying.
For the geometric series $5+15+45+135+405,$ identify $a$ , $r$ , and $n$ , then compute the sum $S_n$ .
Solve the system using the inverse of the coefficient matrix. Write $AX=B$ and compute $X=A^{-1}B$ . $\begin{cases} 4x - y = 9 \\ 2x + y = 7 \end{cases}$ What is $(x,y)$ ?
An investment grows according to $A(t)=A_0(1.09)^t$ , where $t$ is in years. Which expression is equivalent and uses exponent properties to show an approximate monthly growth model of the form $(\text{monthly factor})^{12t}$ ? (You may use $1.09^{1/12}\approx 1.0072$ .)
In the expression $5(x - 1)^2 + 2x(x + 3) - 4$ , identify all the terms. (Do not expand; terms are separated by $+$ or $-$ at the top level.)
Points $z_1=-5+0i$ and $z_2=1+8i$ lie on the complex plane. Using the midpoint formula $\frac{z_1 + z_2}{2} = \left( \frac{a + c}{2} \right) + \left( \frac{b + d}{2} \right)i$ , what is the midpoint of the segment joining them?
Use synthetic division to divide $P(x)=2x^4-3x^3+x^2-4x+6$ by $(x-1).$ Synthetic division is faster than long division here because the divisor is linear of the form $(x-c)$ . What are the quotient and remainder? (Remainder $=P(1)$ .)
Find the modulus $r$ and an argument $\theta$ (in radians) for $z=-1+i$ , and write $z$ in 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 from the positive real axis.)
The identity $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ can be used to factor certain expressions. For which value of $k$ does the expression $8x^3 - k$ factor as $(2x - 3)(4x^2 + 6x + 9)$ ?
Interpret by chunking to reveal multiplicative structure: $\frac{P(1+r)^n}{1+r}.$ Here $P$ is principal and $r,n$ are constants with respect to $P$ . Which statement correctly treats $(1+r)^n$ and $(1+r)$ as entities and identifies what does not depend on $P$ ?
Consider the polynomial $f(x) = 6x^4 - 7x^3 + 2x^2 - 8x + 12$ . If $p$ is a rational zero of $f(x)$ , which of the following statements must be true about the numerator and denominator of $p$ when written in lowest terms as $\frac{a}{b}$ ?
For $g(x)=a\cdot b^x$ , consider equal intervals of length $h=3$ . Which expression correctly gives the constant factor $\dfrac{g(x+3)}{g(x)}$ , and what does that say about growth over every 3-unit step?
An online store's shipping cost is modeled by $S = 1.25w + 8$ , where $S$ is in dollars and $w$ is the package weight in pounds. In the function $S = 1.25w + 8$ , interpret the slope $1.25$ .
The pH of a solution is given by $\text{pH}=-\log_{10}([H^+])$ , where $[H^+]$ is the hydrogen ion concentration in moles per liter. A solution has pH 4.2. Create an equation and solve for $[H^+]$ . Give your answer in scientific notation and interpret it in context.
If $g(x) = 2x^3 - 1$ and $g^{-1}(x)$ represents the inverse function of $g(x)$ , which equation could be used to find $g^{-1}(15)$ ?
Express $\frac{3x^3-2x^2+4}{x+1}$ as $q(x)+\frac{r(x)}{x+1}$ using polynomial division, with $\deg(r)<1$ .
What is the LCD (least common denominator) for the rational expressions $\frac{3}{x^2-4} \quad\text{and}\quad \frac{2}{x^2+4x+4}?$ (Factor denominators first.)
Use structure to factor. The expression $x^3-27$ is a difference of cubes. Which factorization is correct?
Function $f$ is defined by $f(x) = -2x^2 + 8x - 3$ . Function $g$ is represented by the table. Which statement comparing the maximum values of these functions is correct?
What are the factors of the term $6x^2y$ ? (Factors are parts multiplied together.)

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