Algebra II Final Exam Test
Algebra II Final Exam
Spring 2026 Semester
Name:
Period:
Solve each equation below.
1. \(x^2 - 5x + 6 = 0\)
A) \(x = 2, x = 3\)
B) \(x = -2, x = -3\)
C) \(x = 6, x = -1\)
D) \(x = 1, x = -6\)
2. \(2x^2 + 7x + 3 = 0\)
A) \(x = -1, x = -3\)
B) \(x = -\frac{1}{2}, x = -3\)
C) \(x = \frac{1}{2}, x = 3\)
D) \(x = -\frac{3}{2}, x = -1\)
3. \(2x^2 - 4x - 1 = 0\)
A) \(x = \frac{2 \pm \sqrt{6}}{2}\)
B) \(x = 2 \pm \sqrt{6}\)
C) \(x = 1 \pm \sqrt{6}\)
D) \(x = \frac{4 \pm \sqrt{2}}{4}\)
4. \(x^2 + 6x + 7 = 0\)
A) \(x = -3 \pm \sqrt{2}\)
B) \(x = -6 \pm \sqrt{2}\)
C) \(x = -3 \pm 2\sqrt{2}\)
D) \(x = 3 \pm \sqrt{2}\)
5. \(x^2 - 9 = 0\)
A) \(x = 3\) only
B) \(x = 9, x = -9\)
C) \(x = 3, x = -3\)
D) \(x = \sqrt{3}, x = -\sqrt{3}\)
6. \(3x^2 - 2x - 2 = 0\)
A) \(x = \frac{1 \pm \sqrt{7}}{3}\)
B) \(x = \frac{2 \pm \sqrt{7}}{3}\)
C) \(x = \frac{1 \pm 2\sqrt{7}}{3}\)
D) \(x = \frac{1 \pm \sqrt{5}}{3}\)
For questions 7 - 12, use \(f(x) = 2x + 3\) and \(g(x) = x^2 - 1\).
7. Find \((f + g)(x)\).
A) \(x^2 + 2x + 2\)
B) \(x^2 + 2x + 4\)
C) \(2x^2 + 2\)
D) \(3x^2 + 2\)
8. Find \((g - f)(x)\).
A) \(x^2 - 2x + 2\)
B) \(x^2 - 2x - 4\)
C) \(-x^2 + 2x + 4\)
D) \(x^2 + 2x - 2\)
9. Find \((f \cdot g)(x)\).
A) \(2x^3 - 3\)
B) \(2x^3 + 3x^2 - 2x - 3\)
C) \(2x^3 - 2x + 3\)
D) \(x^2 + 2x + 2\)
10. Find \(f(g(x))\).
A) \(2x^2 + 1\)
B) \(2x^2 + 2\)
C) \(4x^2 + 12x + 8\)
D) \(2x^2 - 1\)
11. Find \(g(f(x))\).
A) \(4x^2 + 9\)
B) \(2x^2 + 5\)
C) \(4x^2 + 12x + 8\)
D) \(4x^2 + 12x + 9\)
12. Find \((g \circ f)(-2)\).
A) \(0\)
B) \(8\)
C) \(12\)
D) \(-1\)
Find the inverse of each function below.
13. \(f(x) = 3x - 5\)
A) \(f^{-1}(x) = 5x - 3\)
B) \(f^{-1}(x) = \frac{x-5}{3}\)
C) \(f^{-1}(x) = \frac{x+5}{3}\)
D) \(f^{-1}(x) = \frac{x}{3} + 5\)
14. \(f(x) = \frac{1}{2}x + 4\)
A) \(f^{-1}(x) = 2x - 4\)
B) \(f^{-1}(x) = 2x - 8\)
C) \(f^{-1}(x) = 2x + 8\)
D) \(f^{-1}(x) = \frac{1}{2}x - 4\)
15. \(f(x) = x^3 + 2\)
A) \(f^{-1}(x) = \sqrt[3]{x} - 2\)
B) \(f^{-1}(x) = \sqrt[3]{x+2}\)
C) \(f^{-1}(x) = \sqrt[3]{x-2}\)
D) \(f^{-1}(x) = x^3 - 2\)
16. \(f(x) = \sqrt{x-1}\) for \(x \geq 1\)
A) \(f^{-1}(x) = x^2 + 1\)
B) \(f^{-1}(x) = x^2 - 1\)
C) \(f^{-1}(x) = (x+1)^2\)
D) \(f^{-1}(x) = \sqrt{x+1}\)
17. \(f(x) = \frac{x-4}{2}\)
A) \(f^{-1}(x) = 2x - 4\)
B) \(f^{-1}(x) = 2x + 4\)
C) \(f^{-1}(x) = 4x + 2\)
D) \(f^{-1}(x) = \frac{2}{x-4}\)
18. \(f(x) = 2x^3\)
A) \(f^{-1}(x) = \sqrt[3]{2x}\)
B) \(f^{-1}(x) = \frac{\sqrt[3]{x}}{2}\)
C) \(f^{-1}(x) = \sqrt[3]{\frac{x}{2}}\)
D) \(f^{-1}(x) = \frac{x^3}{2}\)
Simplify each expression.
19. \((3x^2 y^3)^2\)
A) \(6x^4 y^6\)
B) \(9x^4 y^5\)
C) \(9x^4 y^6\)
D) \(9x^2 y^3\)
20. \((2a^2 b) \cdot (-4 a b^4)\)
A) \(-8 a^2 b^4\)
B) \(-8 a^3 b^5\)
C) \(8 a^3 b^5\)
D) \(-2 a^3 b^5\)
21. \(\frac{12 x^5 y^2}{3 x^2 y^4}\)
A) \(4 x^3 y^2\)
B) \(9 x^3 y^{-2}\)
C) \(\frac{4x^3}{y^2}\)
D) \(\frac{4x^7}{y^6}\)
22. \((x^{-2} y^3)^{-1}\)
A) \(\frac{x^2}{y^3}\)
B) \(\frac{y^3}{x^2}\)
C) \(x^2 y^3\)
D) \(\frac{1}{x^2 y^3}\)
23. \((5 a^0 b^4)^2\)
A) \(10 b^8\)
B) \(25 a^0 b^6\)
C) \(25 b^8\)
D) \(0\)
24. \(\frac{(2 m^2 n)^3}{4 m n^5}\)
A) \(\frac{2 m^5}{n^2}\)
B) \(\frac{2 m^6}{n^2}\)
C) \(\frac{6 m^5}{n^2}\)
D) \(2 m^5 n^2\)
Divide.
25. \((x^2 + 7x + 12) \div (x + 3)\)
A) \(x + 4\)
B) \(x + 3\)
C) \(x - 4\)
D) \(x^2 + 4\)
26. \((x^2 - 5x + 6) \div (x - 2)\)
A) \(x + 3\)
B) \(x - 3\)
C) \(x - 2\)
D) \(x + 2\)
27. \((2x^3 - 5x^2 - 8x + 15) \div (x - 3)\)
A) \(2x^2 + x - 5\)
B) \(2x^2 - x + 5\)
C) \(2x^2 + 11x + 25\)
D) \(2x^2 - 11x - 41\)
28. \((4x^2 - 8x + 3) \div (2x - 1)\)
A) \(2x - 3\)
B) \(2x + 3\)
C) \(2x - 1\)
D) \(x - 3\)
29. \((3x^3 + x^2 + 4x - 1) \div (x^2 + 2)\)
A) \(3x + 1 + \frac{-2x - 3}{x^2 + 2}\)
B) \(3x - 1 + \frac{2x + 1}{x^2 + 2}\)
C) \(3x + 1 + \frac{-3}{x^2 + 2}\)
D) \(3x + 1 + \frac{-2x - 1}{x^2 + 2}\)
30. \((2x^2 + 5x - 1) \div (x + 2)\)
A) \(2x + 1 + \frac{1}{x+2}\)
B) \(2x + 9 + \frac{17}{x+2}\)
C) \(2x + 1 - \frac{3}{x+2}\)
D) \(2x - 1 + \frac{1}{x+2}\)
© 2026 Mathematics Department - Algebra II Spring Final Exam
