Practice Test 74 - Algebra 2

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

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1. Which option correctly applies the identity $(a+b)^2 = a^2 + 2ab + b^2$ with complex numbers $a = i$ and $b = 2 - i$ to compute $(i + (2 - i))^2$?
2. Let $z = -5 + 12i$. On the complex plane, $|z|$ is the distance from the origin to the point $(-5,12)$. What is $|z|$?
3. Explain why $f(x)=x^2$ (domain: all real numbers) needs a domain restriction to have an inverse by using the horizontal line test. Which statement correctly describes the issue?
4. What is the coefficient of $x^2$ in the polynomial $3x^3 - 7x^2 + 2x - 9$?​
5. Find the linear function $f(x)=mx+b$ with slope $m=-3$ that passes through the point $(2,5)$. Which function is correct?
6. Let $z = -3 + 8i$. Which value equals $z\overline{z}$? (Recall $z\overline{z}=a^2+b^2$, the modulus squared.)
7. Two complex numbers $u = 1 + i$ and $v = \sqrt{3} - i$ are multiplied together. If this multiplication is interpreted geometrically as a composition of scaling and rotation, what is the total angle of rotation applied to $u$?
8. A quadratic with real coefficients (of the form $ax^2+bx+c=0$) always has exactly 2 complex solutions counting multiplicity. Also, if one non-real complex solution is $a+bi$, the other must be its conjugate $a-bi$. If $x^2-4x+13=0$ has a solution $2+3i$, what is the other solution?
9. Factor the expression completely by recognizing repeated difference-of-squares structure: $x^6 - 64$
10. Apply the Rational Zeros Theorem to list all possible rational zeros of $P(x)=4x^3-3x^2-10x+5.$ (Give the full candidate list in lowest terms.)
11. What is the recursive definition for the arithmetic sequence $12,\ 7,\ 2,\ -3,\dots$ (integer domain $n\ge 1$)?
12. A student claims that since $g(x) = 0.5^x$ decreases while $f(x) = x^2$ increases, the quadratic function will always be larger than the exponential function for positive $x$ values. Evaluate this claim and identify the error in reasoning.
13. A temperature function $T(h) = 70 - 0.004h$ gives temperature in °F at altitude $h$ feet above sea level. Aircraft regulations require temperatures above 32°F. Based on the data table shown, what domain restriction ensures regulatory compliance?
14. A quadratic equation can be written as $(x - 4)^2 = 25$. Which of the following represents all solutions when this equation is solved using a different method?
15. A quadratic polynomial $g(x)$ has integer coefficients and roots $\alpha = \frac{1+i\sqrt{7}}{2}$ and $\beta = \frac{1-i\sqrt{7}}{2}$. What is the minimum possible value of $|g(1)|$?
16. A researcher observes that a bacterial culture triples every 4 hours, while a plant grows 2.5 cm taller every 4 hours. If the bacteria count starts at 100 and the plant starts at 8 cm tall, which statement correctly compares their growth after 12 hours?
17. Solve for $t$: $15\cdot e^{t/4}=120$. Express $t$ exactly using a logarithm and then approximate using a calculator.
18. Use logarithm properties to evaluate without a calculator: $\log_3\left(\frac{81}{9}\right)$
19. Recognize a pattern and factor completely using polynomial identities: $x^4-16.$
20. Use Pascal's Triangle to expand $(x+2)^4.$

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