Practice Test 53 - Algebra 2

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

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1. Let $P(x) = 3x^3 - 2x^2 + x - 5$ and $Q(x) = x^3 + 4x^2 - 3x + 1$. If $R(x) = P(x) - Q(x) + 2P(x)$, what is the coefficient of $x^2$ in $R(x)$?
2. Find the modulus $r$ and a correct argument $\theta$ (in degrees) for $-2+2i$, then write it 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 counterclockwise from the positive real axis.)
3. Determine if $x=-1$ is a zero of $p(x)=x^3+2x^2-x-2.$ (Use the fact that $x=a$ is a zero iff $p(a)=0$.)
4. The rational function $g(x) = \frac{2x^3 - 8x}{x^2 + x - 6}$ can be simplified by factoring. What is the equation of the vertical asymptote after simplification?
5. Sketch the logarithmic function $k(x)=\ln(x+2)$ showing the vertical asymptote and the x-intercept.
6. You deposit $1200 into an account that earns 3.8% interest compounded annually. Create an exponential model for the account balance after $t$ years, then determine how long it will take for the balance to double. Round to the nearest tenth of a year and interpret your result.
7. A quantity $y$ is recorded for integer values of $x$: $x$: 1, 2, 3, 4 $y$: 30, 36, 43.2, 51.84 From the table, what is the percent growth rate per 1-unit increase in $x$?
8. For the polynomial $t(x)=x^4-5x^2,$ identify all real zeros and state the end behavior.
9. A streaming service charges $9 per month plus $2 for each movie rental. Let $m$ be the number of months and $r$ be the number of rentals in that time. Write an equation for the total cost $T$ (dollars) in terms of $m$ and $r$. Then, to graph in 2D, suppose $m=3$ months—what equation should you graph for $T$ as a function of $r$, and what axes labels and scale are reasonable for $0 \le r \le 20$?
10. What are the period and midline of the trigonometric function $q(x)=3\cos\left(\frac{x}{2}\right)-4$?
11. For $f(x)=\frac{(x-1)(x+2)}{(x-1)(x-4)},$ identify any hole(s), vertical asymptote(s), zero(s), and the horizontal asymptote.
12. Solve the quadratic equation by factoring: $x^2 - 9x + 20 = 0$
13. A polynomial $P(x)$ has degree 4, and when $P(x)$ is added to the polynomial $3x^3 - 2x^2 + x - 5$, the result is a polynomial of degree 3. Which statement about $P(x)$ must be true?
14. Show that $x=2$ and $x=-3$ are zeros of $f(x)=x^2+x-6$ by factoring, then state the x-intercepts.
15. A line and a parabola intersect where their equations are both true. Solve the system algebraically. $\begin{cases} y = x + 1 \\ y = x^2 - 2x + 1 \end{cases}$ What are the intersection point(s)?
16. Simplify and write the result in standard form (this uses polynomial operations and demonstrates closure under $+$, $-$, and $\times$): $(x + 2)^2 - (x - 1)(x + 3)$
17. Expand $(x-2)^4$ using the Binomial Theorem (Pascal's Triangle row 4: $1,4,6,4,1$).
18. A delivery app models the total pay (in dollars) for completing $n$ deliveries as $P(n)=7n+15$. A driver can complete at most 25 deliveries in a shift. What values of $n$ are appropriate for this context?
19. Use exponent structure to rewrite the expression as a single power of $2$: $8^{2x}$
20. Multiply the polynomials (distribute each term). The result is a polynomial by closure under multiplication: $(x^2+2x-1)\times(x^2-x+3).$​

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